Mathilde Couteaudier
email: mathilde.couteaudier@hotmail.fr

Diego Calzada-Fraile
email: diego.calzadafraile@gmail.com

Thomas Montange
email: thomas.montange@pasteur.fr

Antoine Gessain
email: antoine.gessain@pasteur.fr

Florence Buseyne
email: florence.buseyne@pasteur.fr

Inhibitors of the Interferon Response Increase the Replication of Gorilla Simian Foamy Viruses

Keywords: foamy virus, spumaretroviruses, zoonosis, immune response, interferon, virus replication

Simian foamy viruses (SFVs) are complex retroviruses that are widespread throughout nonhuman primates. SFVs can also be transmitted to humans, mostly through bites. We previously observed that primary zoonotic gorilla SFV strains grow much more slowly than laboratory-adapted chimpanzee strains. Here, we tested the hypothesis that the growth of SFV is limited by interferon (IFN) using inhibitors of cellular pathways involved in the induction or action of type I IFN. Inhibitors of JAK1/2 (Ruxolitinib) and TBK-1 (BX795) led to a 2-to 4-fold higher percentage of cells infected with zoonotic gorilla SFVs but did not affect the replication of laboratory-adapted chimpanzee SFVs.

IKK2 inhibitors (TPCA-1 and BMS345541) had no effect on any of the SFV strains. In conclusion, the addition of molecules that inhibit the type I IFN response to the culture medium can be used as a simple and efficient method to enhance the replication of zoonotic gorilla SFVs.

Introduction

Simian foamy viruses (SFVs) are complex retroviruses that are widespread throughout nonhuman primates (NHPs) (reviewed in [START_REF] Gessain | HTLV-3/4 and simian foamy retroviruses in humans: Discovery, epidemiology, cross-species transmission and molecular virology[END_REF]). SFVs can also be transmitted to humans through bites, establishing a persistent infection [START_REF] Gessain | HTLV-3/4 and simian foamy retroviruses in humans: Discovery, epidemiology, cross-species transmission and molecular virology[END_REF][START_REF] Betsem | Frequent and recent human acquisition of simian foamy viruses through apes' bites in Central Africa[END_REF][START_REF] Rua | Origin, evolution and innate immune control of simian foamy viruses in humans[END_REF][START_REF] Filippone | A severe bite from a nonhuman primate is a major risk factor for HTLV-1 infection in hunters from Central Africa[END_REF][START_REF] Pinto-Santini | Foamy virus zoonotic infections[END_REF]. To date, neither clinical effects nor secondary transmission between humans have been reported [START_REF] Pinto-Santini | Foamy virus zoonotic infections[END_REF][START_REF] Buseyne | Clinical signs and blood test results among humans infected with zoonotic simian foamy virus: a case-control study[END_REF]. Replication-competent SFVs have been isolated from blood cells years or even decades after infection [START_REF] Schweizer | Simian foamy virus isolated from an accidentally infected human individual[END_REF][START_REF] Boneva | Clinical and virological characterization of persistent human infection with simian foamy viruses[END_REF][START_REF] Rua | Genetic characterization of simian foamy viruses infecting humans[END_REF] and cell-associated SFV DNA has been detected in human blood and buccal samples [START_REF] Betsem | Frequent and recent human acquisition of simian foamy viruses through apes' bites in Central Africa[END_REF][START_REF] Boneva | Clinical and virological characterization of persistent human infection with simian foamy viruses[END_REF][START_REF] Calattini | Simian foamy virus transmission from apes to humans, rural Cameroon[END_REF][START_REF] Jones-Engel | Diverse contexts of zoonotic transmission of simian foamy viruses in Asia[END_REF][START_REF] Switzer | Novel simian foamy virus infections from multiple monkey species in women from the Democratic Republic of Congo[END_REF][START_REF] Engel | Zoonotic simian foamy virus in Bangladesh reflects diverse patterns of transmission and co-infection[END_REF][START_REF] Rua | In vivo cellular tropism of gorilla simian foamy virus in blood of infected humans[END_REF]. An understanding of the persistence of SFV in humans and its consequences requires in vitro experiments performed with primary strains. Indeed, strains may differ in cell tropism and susceptibility to inhibition by innate and adaptive immune effector molecules and cells [START_REF] Schweizer | Simian foamy virus isolated from an accidentally infected human individual[END_REF][START_REF] Rua | Genetic characterization of simian foamy viruses infecting humans[END_REF][START_REF] Rua | Innate sensing of foamy viruses by human hematopoietic cells[END_REF][START_REF] Lambert | Potent neutralizing antibodies in humans infected with zoonotic simian foamy viruses target conserved epitopes located in the dimorphic domain of the surface envelope protein[END_REF]. However, repeated cycles of in vitro production may lead to the selection of the fittest viral variants in specific cell-culture conditions and/or the expansion of strains that would have been efficiently repressed in vivo by immune responses. Thus, tropism and susceptibility to interferons (IFNs) or antibodies frequently differ between laboratoryadapted and primary strains of retroviruses [START_REF] Kabat | Differences in CD4 dependence for infectivity of laboratory-adapted and primary patient isolates of human immunodeficiency virus type 1[END_REF][START_REF] Wrin | Adaptation to persistent growth in the H9 cell line renders a primary isolate of human immunodeficiency virus type 1 sensitive to neutralization by vaccine sera[END_REF][START_REF] Cook | Enhanced sensitivity to neutralizing antibodies in a variant of equine infectiousanemia virus is linked to amino-acid substitutions in the surface unit envelope glycoprotein[END_REF][START_REF] Mascola | HIV-1 neutralizing antibodies: understanding nature's pathways[END_REF].

Central African villagers from East and South Cameroon and North Gabon are at high risk for SFV acquisition and approximately 75% of infected individuals carry strains from gorillas [START_REF] Betsem | Frequent and recent human acquisition of simian foamy viruses through apes' bites in Central Africa[END_REF][START_REF] Buseyne | Clinical signs and blood test results among humans infected with zoonotic simian foamy virus: a case-control study[END_REF][START_REF] Calattini | Simian foamy virus transmission from apes to humans, rural Cameroon[END_REF][START_REF] Mouinga-Ondeme | Cross-species transmission of simian foamy virus to humans in rural Gabon, Central Africa[END_REF]. The only gorilla SFV isolated directly from an animal has limited in vitro replication capacity [START_REF] Bieniasz | A comparative study of higher primate foamy viruses, including a new virus from a gorilla[END_REF].

We previously isolated two zoonotic gorilla SFV strains and demonstrated their high genetic similarity with their NHP counterparts [START_REF] Rua | Genetic characterization of simian foamy viruses infecting humans[END_REF]. The sequences of these primary viral strains are identical to those of gene fragments amplified from human PBMC-associated viral DNA [START_REF] Rua | Genetic characterization of simian foamy viruses infecting humans[END_REF]. Next, we performed in vitro infections using viral stocks produced after inoculation with original or firstpassage cell lysates, which are referred to as primary strains [START_REF] Rua | Innate sensing of foamy viruses by human hematopoietic cells[END_REF][START_REF] Lambert | Potent neutralizing antibodies in humans infected with zoonotic simian foamy viruses target conserved epitopes located in the dimorphic domain of the surface envelope protein[END_REF][START_REF] Lambert | A new sensitive indicator cell line reveals crosstransactivation of the viral LTR by gorilla and chimpanzee simian foamy viruses[END_REF]. We observed that primary gorilla SFV grow more slowly than laboratory-adapted chimpanzee SFV on hamster kidney (BHK-21) cells, which are used for their isolation and production [START_REF] Lambert | A new sensitive indicator cell line reveals crosstransactivation of the viral LTR by gorilla and chimpanzee simian foamy viruses[END_REF]. Furthermore, primary gorilla SFV stocks collected from cell cultures displaying an extensive cytopathic effect have 100-fold lower cell-free titers on the GFAB indicator cells in which their autologous LTR directs the expression of a reporter molecule [START_REF] Lambert | A new sensitive indicator cell line reveals crosstransactivation of the viral LTR by gorilla and chimpanzee simian foamy viruses[END_REF].

Among host factors that affect replication efficacy, the induction and action of IFNs are of utmost importance [START_REF] Schneider | Interferon-stimulated genes: a complex beb of host defenses[END_REF][START_REF] Mcnab | Type I interferons in infectious disease[END_REF]. Pattern-recognition receptors recognize viral nucleic acids and initiate a molecular cascade, resulting in IFN regulatory factor (IRF)3, IRF7, and/or nuclear factor-kappa B (NF-κB) translocation to the nucleus and the triggering of IFN gene transcription [START_REF] Schneider | Interferon-stimulated genes: a complex beb of host defenses[END_REF]. In plasmacytoid dendritic cells, SFVs trigger type I IFN production after being sensed by Toll-like receptor 7 [START_REF] Rua | Innate sensing of foamy viruses by human hematopoietic cells[END_REF]. Secreted IFN binds to its cellular receptor and initiates a signaling cascade through the Janus kinase signal transducer and activator of transcription (JAK-STAT) pathway [START_REF] Schneider | Interferon-stimulated genes: a complex beb of host defenses[END_REF][START_REF] Mcnab | Type I interferons in infectious disease[END_REF]. Type I IFNs restrict SFV replication [START_REF] Rua | Innate sensing of foamy viruses by human hematopoietic cells[END_REF][START_REF] Matthes | Basic residues in the foamy virus Gag protein[END_REF][START_REF] Bähr | Interferon but not MxB inhibits foamy retroviruses[END_REF]]. Among IFN-stimulated genes (ISGs), tripartite motif protein 5α, apolipoprotein B mRNA editing catalytic 3G, N-myc interactor, IFN-induced protein 35, and tetherin inhibit the replication of the laboratory-adapted chimpanzee prototype FV strain (CI-PFV) [START_REF] Rua | Origin, evolution and innate immune control of simian foamy viruses in humans[END_REF][START_REF] Lochelt | The antiretroviral activity of APOBEC3 is inhibited by the foamy virus accessory Bet protein[END_REF][START_REF] Russell | Foamy virus Bet proteins function as novel inhibitors of the APOBEC3 family of innate antiretroviral defense factors[END_REF][START_REF] Berka | Early events in foamy virus-host interaction and intracellular trafficking[END_REF]. Systematic screening of ISGs has identified additional CI-PFV inhibitors [START_REF] Kane | Identification of interferon-stimulated genes with antiretroviral activity[END_REF].

The titers of primary gorilla SFV stocks produced on BHK-21 cells are usually low and sometimes insufficient to carry out in vitro infections. Therefore, we tested three strategies to enhance the level and/or speed of primary gorilla SFV replication: the search for sensitive human cell lines, improvement of viral entry, and blockade of the type I IFN response. The third approach was successful, as small molecules that inhibit IFN signaling or the IFN response enhanced the replication of zoonotic gorilla SFVs.

Results

Primary gorilla SFV strains replicate more slowly than laboratory-adapted chimpanzee SFV strains in adherent cell lines Undiluted stocks of primary zoonotic gorilla SFV strains GI-D468 and GII-K74 were used to infect BHK-21 cells. Infection was monitored by flow cytometry at each cell passage, twice per week.

Infected cell frequencies > 1% were first detected at day 10-11 or day 13-14 post-infection (pi) (Figure 1A). The infected cell frequencies increased sharply at the following passage. Then, rapid, massive cell lysis occurred. The kinetics of infection varied between experiments and strains: GI-D468 spread more rapidly than GII-K74 in the first experiment and more slowly in the second (Figure 1A). We monitored the appearance of CPE in human fibroblasts (MRC5), U-87MG, murine mus dunni, and simian IFN-deficient VERO cells before constructing the GFAB indicator cells and using the anti-Env monoclonal antibody. All showed similar or lower susceptibility to gorilla SFV than BHK-21 cells (Lambert, Gouzil and Buseyne, data not shown).

The human HT1080 cell line has been reported to be more susceptible to CI-PFV than BHK-21 cells [START_REF] Plochmann | Heparan sulfate is an attachment factor for foamy virus entry[END_REF]. We thus infected HT1080 cells with GI-D468 and GII-K74. Infection frequencies > 1% were first detected at day 18 (Figure 1A). The infection of HT1080 with gorilla SFV was reproducible (data not shown), but the replication kinetics were variable, with infection frequencies > 1% detected between the second and third week of infection. We used the laboratory-adapted CI-PFV and CII-SFV7 strains as positive controls of infection.

We observed infected cell frequencies > 1% at between 3 and 7 days pi, depending on the experiment and cells, using the same multiplicity of infection (MOI) as for primary gorilla SFV (Figure 1A). In conclusion, primary gorilla SFV strains spread more slowly than the chimpanzee SFV strains in adherent cell lines, even when infections were carried out at the same MOI. In addition, frequencies of infected cells varied between independent experiments at a given timepoint pi.

Human hematopoietic cell lines are poorly susceptible to infection with SFVs

We then tested whether human hematopoietic cell lines are susceptible to gorilla SFV, as we previously reported that human blood lymphoid cells contain SFV DNA [START_REF] Rua | In vivo cellular tropism of gorilla simian foamy virus in blood of infected humans[END_REF]. We first tested the susceptibility of four cell lines using the rapidly replicating CI-PFV strain. Cell pellets were infected with either undiluted or 10-fold-diluted stocks that had a high infectious titer (2 10 6 IU/ml). The corresponding MOI were 0.6 and 0.06. K562 cells were productively infected using a MOI of 0.6, with a frequency of infected cells of 75% at day 23 (Figure 1B). We observed a delayed infection at a MOI of 0.06. HEL 92.1.7, Raji, and Jurkat cells did not express SFV Env. Low frequencies of infection in THP-1 cells were not sustained. We infected K562 cells with undiluted GI-468 (MOI 0.0015), undiluted CI-PFV (MOI 0.6), and CI-PFV diluted to a MOI of 0.0015. At a MOI of 0.0015, we detected > 10% GI-D468-infected cells at day 41 (Figure 1C). After infection with CI-PFV at the high MOI, infected K562 cells were detected at day 7 and massive cell lysis occurred at day 28; no infected cells were detectable upon infection with CI-PFV at a MOI of 0.0015. We attempted to concentrate viral stocks to reach a higher MOI. However, the concentration of SFV particles by iodixanol-gradient ultracentrifugation or centrifugal filters resulted in reduced infectivity and lower infectious titers on GFAB cells (data not shown). In conclusion, although hematopoietic K562 cells can support primary gorilla SFV, several weeks of culture were required before the detection of infected cells due to the low infectious titers of the viral stocks. Thus, we did not further investigate the susceptibility of various cell lines to gorilla SFV, but sought treatment of the cells that could improve the efficacy of viral entry and spread.

Procedures targeting the early steps of the replication cycle do not enhance infection with gorilla SFV

SFV Env-mediated fusion is inducible by short exposure to acidic pH when expressed at the cell surface [START_REF] Picard-Maureau | Foamy virus envelope glycoprotein-mediated entry involves a pH-dependent fusion process[END_REF]. The fusion process is much slower for macaque SFV Env than CI-PFV Env [START_REF] Stirnnagel | Differential pH-dependent cellular uptake pathways among foamy viruses elucidated using dual-colored fluorescent particles[END_REF]. We thus tested whether changing the pH could enhance infection with viral particles. GFAB cells were infected for 2 h, exposed to pH 7 or pH 5.5 for 1 or 15 min, and cultured for 72 h. Exposure to acidic pH had no effect on the number of infected GI-D468 cells and decreased the number of those infected with CI-PFV (Figure 2A). Cationic polymers can also increase the adsorption of virions on target cell membranes, leading to the enhancement of infection of certain retroviruses [START_REF] Toyoshima | Enhancement and inhibition of avian sarcoma viruses by polycations and polyanions[END_REF]. We infected GFAB cells with GI-D468 or CI-PFV in the presence of polyethylenimine, polybrene, or DEAE dextran at concentrations ranging from 0.5 to 8 µg/mL. Cationic polymers did not increase the frequency of infection with GI-D468 or CI-PFV, and even decreased infectivity at the highest concentrations (Figure 2C).

IFN inhibitors enhance infection with gorilla SFVs

We addressed whether SFV replication is sensitive to type I IFN-mediated restriction by adding small molecules that inhibit kinases involved in IFN induction (TBK-1 and IKK2) or signaling (JAK1/2) [START_REF] Clark | Use of the pharmacological inhibitor BX795 to study the regulation and physiological roles of TBK1 and IkappaB kinase epsilon: a distinct upstream kinase mediates Ser-172 phosphorylation and activation[END_REF][START_REF] Stewart | Inhibitors of the interferon response enhance virus replication in vitro[END_REF][START_REF] Cataldi | Breaking resistance of pancreatic cancer cells to an attenuated vesicular stomatitis virus through a novel activity of IKK inhibitor TPCA-1[END_REF]. Inhibitors of JAK1/2 (Ruxolitinib), TBK-1 (BX795), or IKK2 (TPCA-1 and BMS345541) were added to cultures 4 h before GFAB cell infection and maintained during the 72 h-long culture.

Ruxolitinib and BX795 increased the number of GFAB cells infected with GI-D468 and GII-BAK74 (t test, P < 0.05, Figure 3A-B). There was no additive effect if both molecules were mixed (data not shown). At the highest concentrations tested, there was a cytotoxic effect that was more evident for BX795. Neither of the two IKK2 inhibitors affected GFAB cell infection (Figure 3C-D). The CI-PFV strain, used as positive control of infection, was not affected by any of these four molecules (Figure 3A-D). Thus, blocking IFN signaling through JAK1/2 or the TBK-1-dependent IFN response increased GFAB cell infection with zoonotic gorilla SFVs.

We then tested the effect of ruxolitinib on several cycles of SFV replication and followed Env expression by flow cytometry. Cells were stained after the appearance of syncytia. Addition of ruxolitinib increased the number of GFAB cells infected with GI-D468 and GII-K74 (P = 0.04 and P = 0.03, respectively, Figure 4). The percentage of Env + HT1080 increased by at least 2 fold with ruxolitinib treatment (data not shwon). However, the percentage of infected cells was unrelated to the observed cytopathic effect. The loss of infected HT1080 cells during cell harvest and staining is likely and probably resulted in an understimation of the SFV replication. There was no effect of ruxolitinib on CI-PFV replication in GFAB (Figure 4) or HT1080 cells (data not shown) under the same conditions of infection and culture (data not shown). In conclusion, we confirmed the enhancing effect of JAK1/2 inhibition on the spread of zoonotic gorilla SFV in infected cultures.

Discussion

We sought to increase the in vitro replication of primary gorilla SFV strains and found that inhibition of the type I IFN response is an efficient way to achieve this goal. Indeed, the supplementation of culture medium with JAK1/2 or TBK-1 inhibitors is a simple and broadly applicable method that works well for a variety of viruses [START_REF] Stewart | Inhibitors of the interferon response enhance virus replication in vitro[END_REF][START_REF] Cataldi | Breaking resistance of pancreatic cancer cells to an attenuated vesicular stomatitis virus through a novel activity of IKK inhibitor TPCA-1[END_REF][START_REF] Ma | NLRX1 negatively modulates type I IFN to facilitate KSHV reactivation from latency[END_REF], including retroviruses [START_REF] Decalf | Sensing of HIV-1 entry triggers a type I interferon response in human primary macrophages[END_REF].

The blockade of IFN signaling by JAK1/2 and the TBK-1-mediated IFN response increased gorilla SFV replication. In contrast, the blockade of another component of the IFN response, IKK2, had no effect on in vitro SFV growth. Conclusions on the cellular pathways that affect the spread of SFV based on the use of pharmacological inhibitors must be made with caution. First, the pleiotropic JAK/STAT signaling pathway regulates the response to not only IFN, but also to cytokines, and several cellular processes, including cell proliferation [START_REF] Rawlings | The JAK/STAT signaling pathway[END_REF]. Indeed, two opposite effects could be expected from JAK1/2 inhibition: the promotion of viral replication through the inhibition of IFN signaling and the reduction of viral spread through the blockade of mitosis, which is required for SFV provirus integration [START_REF] Trobridge | Cell cycle requirements for transduction by foamy virus vectors compared to those of oncovirus and lentivirus vectors[END_REF]. Furthermore, some of the inhibitors we used target several molecules. BX795 blocks TBK-1 and Akt [START_REF] Jaishankar | An off-target effect of BX795 blocks herpes simplex virus type 1 infection of the eye[END_REF], whereas TPCA-1 inhibits both IKKs and STAT3 [START_REF] Nan | TPCA-1 is a direct dual inhibitor of STAT3 and NF-kappaB and regresses mutant EGFRassociated human non-small cell lung cancers[END_REF].

Our primary aim was to enhance the in vitro growth of primary gorilla SFV. We thus included the laboratory-adapted chimpanzee CI-PFV strain as a positive control of infection in every experiment. Its susceptibility to IFN blockade differed from that of gorilla SFV strains. Indeed, CI-PFV infection was not enhanced by ruxolitinib or BX795. Several nonexclusive mechanisms could account for such a strain-specific effect. As discussed above, pharmacological inhibitors may block several pathways and/or have several targets with opposite effects. For example, JAK/STAT inhibitors promote the early steps of replication of HIV-1 Env mutant strains in T-cell lines but have no effect on wildtype virus [START_REF] Quan | JAK-STAT Signaling Pathways and Inhibitors Affect Reversion of Envelope-Mutated HIV-1[END_REF]. However, the same molecules decrease wildtype HIV replication in blood primary T lymphocytes [START_REF] Gavegnano | Ruxolitinib and tofacitinib are potent and selective inhibitors of HIV-1 replication and virus reactivation in vitro[END_REF]. Similarly, certain strain-specific properties may underlie SFV susceptibility to IFN blockade. Few side-by-side comparisons of SFV strains have been carried out.

One showed that CI-PFV entry is pH independent, in contrast to several other SFV strains [START_REF] Picard-Maureau | Foamy virus envelope glycoprotein-mediated entry involves a pH-dependent fusion process[END_REF]; this step of viral replication is affected by the IFN response [START_REF] Jakobsdottir | On the Whereabouts of HIV-1 Cellular Entry and Its Fusion Ports[END_REF]. Likewise, the rapid and high-level replication of CI-PFV may saturate antiviral molecules targeted by the drugs we used and explain the strain-specific effect of IFN blockade. In addition, CI-PFV stocks may contain fewer noninfectious virions that can induce the IFN response than the GI-D468 and GII-K74 stocks. Finally, the effect of IFN blockade may depend on the cell type. Indeed, CI-PFV is unable to induce IFN production in nonhematopoietic human epithelial cells [START_REF] Rhodes-Feuillette | Studies on in vitro interferon induction capacity and interferon sensitivity of simian foamy viruses[END_REF][START_REF] Sabile | In vitro studies on interferon-inducing capacity and sensitivity to IFN of human foamy virus[END_REF], whereas it efficiently induces IFN-α production by human plasmacytoid dendritic cells [START_REF] Rua | Innate sensing of foamy viruses by human hematopoietic cells[END_REF].

We tested the susceptibility of several human cell lines to primary gorilla SFVs. No gorilla cell lines or samples are available to test the replication kinetics of gorilla SFV in their natural host cells.

Here, we focused on human cells because SFV infection in humans is our major research theme and because more cell biology and immunology reagents are available for their study. We observed that CI-PFV replicates faster in HT1080 than BHK-21 cells, as reported by others [START_REF] Plochmann | Heparan sulfate is an attachment factor for foamy virus entry[END_REF]. HT1080 cells were susceptible to both GI-D468 and GII-K74, but both gorilla SFVs had similar replication kinetics in HT1080 and BHK-21 cells. Overall, we observed the slow/low replication kinetics of gorilla SFV in both the hamster cells used for their isolation and human fibroblasts, described to have the highest susceptibility to CI-PFV [START_REF] Plochmann | Heparan sulfate is an attachment factor for foamy virus entry[END_REF].

We tested several classical methods to enhance the early steps of SFV infection. The use of polycations to reduce surface charge had no impact on gorilla SFV, as reported by others for PFV [START_REF] Loh | Replication of human syncytium-forming virus in human cells: effect of certain biological factors and selective chemicals[END_REF][START_REF] Hill | Properties of human foamy virus relevant to its development as a vector for gene therapy[END_REF]. The processes that regulate the fusogenic activity of SFV Env and Env in various other viral species are not fully understood [START_REF] Picard-Maureau | Foamy virus envelope glycoprotein-mediated entry involves a pH-dependent fusion process[END_REF]. Although exposure to a pulse of acidic pH induced fusion of gorilla SFV Env expressed at the cell surface, as described in [START_REF] Picard-Maureau | Foamy virus envelope glycoprotein-mediated entry involves a pH-dependent fusion process[END_REF], we observed no effect on SFV Env exposed at the surface of viral particles. We hypothesized that gorilla SFV Env fusion is a slow process, as described for macaque SFV Env [START_REF] Stirnnagel | Differential pH-dependent cellular uptake pathways among foamy viruses elucidated using dual-colored fluorescent particles[END_REF], and exposed cells to low pH after incubation with the viral inoculum. This timing might be suboptimal if most of the particles have already been internalized and are unaffected by the change in pH of the culture medium. Overall, our data argue against a restriction of gorilla SFV replication at the Env-mediated early steps of the viral cycle.

We aimed to avoid the selection of fast-growing laboratory-adapted gorilla SFV strains for performing in vitro infections. We show here that the presence of ruxolitinib during the production of gorilla SFV stocks is a simple way to overcome the slow/low in vitro replication of these strains.

At first glance, blocking restriction mechanisms to enhance SFV growth does not appear to fit our initial goal, because viral particles produced in ruxolitinib-treated cells may differ from those produced in untreated cells, with active innate sensing and an IFN response. It will thus be critical to consider the final experimental outcome before producing viral stocks by IFN blockade. In addition, we propose the use of ruxolitinib for a single round of virus amplification, starting from original material for each new round of production to limit the selection of fitter viral variants over serial passages.

Although we focused on the production of primary SFV stocks, our results may be relevant for the isolation of new replicating SFV strains. Our current procedure relies on the stimulation of human peripheral blood cells to induce viral replication and coculture with susceptible cells from a nonhuman species to avoid inhibition by human IFN-γ and other soluble mediators produced by activated PBMCs [START_REF] Rua | Genetic characterization of simian foamy viruses infecting humans[END_REF][START_REF] Calattini | Simian foamy virus transmission from apes to humans, rural Cameroon[END_REF][START_REF] Falcone | Gamma interferon is a major suppressive factor produced by activated human peripheral blood lymphocytes that is able to inhibit foamy virus-induced cytopathic effects[END_REF]. The use of small molecule inhibitors of the type I IFN response may be useful for improving the isolation of new primary SFV, as demonstrated by the inhibition of IFN-γ [START_REF] Falcone | Gamma interferon is a major suppressive factor produced by activated human peripheral blood lymphocytes that is able to inhibit foamy virus-induced cytopathic effects[END_REF].

In conclusion, we show that the addition of JAK1/2 or TBK-1 inhibitors to culture medium is a simple and efficient procedure to enhance the growth of slow/low primary gorilla SFV isolates. The same inhibitors did not enhance replication of the laboratory-adapted CI-PFV, showing the utility of primary SFV strains for the study of innate sensing. Our results should be useful for researchers in the field of foamy viruses.

Materials and Methods

Cells K562 (ECACC 89121407, human erythroleukemia cells) and BHK-21 (ATCC-CLL-10, hamster kidney fibroblasts) cells were obtained from colleagues at the Institut Pasteur. BHK-21-derived gorilla foamy virus-activated β-galactosidase (GFAB) cells were generated in our laboratory [START_REF] Lambert | A new sensitive indicator cell line reveals crosstransactivation of the viral LTR by gorilla and chimpanzee simian foamy viruses[END_REF].

Other cell lines were purchased from LGC standards or Sigma-Aldrich and stocks were produced after less than 10 passages. The cells were screened for mycoplasma infection every 10 passages with a bioluminescent assay (#LT07-418, Lonza). The HT1080 cells (ECACC 85111505, human fibrosarcoma) were cultivated in Eagle's Minimum Essential Medium with Earle's Balanced Salts and L-Glutamine (EMEM-EBSS, Lonza) supplemented with 10% fetal bovine serum (FBS, PAA Laboratories) and 1% nonessential amino acids (NEAA, Invitrogen). The K562, HEL 92.1.7 (ATCC-TIB-180, human erythroleukemia cells), THP-1 (ECACC 88081201, human monocytic leukemia cells), Raji (ATCC-CCL-86, human B lymphoma cells), and Jurkat (ATCC-TIB-152, human T leukemia cells) cell lines were grown in RPMI medium containing GlutaMAX I (Invitrogen) supplemented with 10% FBS. BHK-21 and GFAB cells were cultivated in Dulbeco's modified Eagle's medium (DMEM-GlutaMAX I, Invitrogen) supplemented with 5% FBS. G418 (300 µg/ml, Sigma-Aldrich) was added to the GFAB cultures.

Viruses

SFV strains consisted of the primary zoonotic gorilla SFVs, SFVggo_huBAD468 (GI-D468) and SFVggo_huBAK74 (GII-K74) [START_REF] Rua | Genetic characterization of simian foamy viruses infecting humans[END_REF], and the laboratory-adapted chimpanzee SFVs, SFVpsc_huPFV (CI-PFV) and SFVpve_Pan2 (CII-SFV7) [START_REF] Herchenroder | Twelfth international foamy virus conference-meeting report[END_REF]. GI-D468 and GII-K74 viral stocks were produced by infecting BHK-21 cells with original or first-passage cell lysates. CI-PFV and CII-SFV7 viral stocks were obtained by a single round of infection with aliquots obtained from A. Saib and A. Rethwilm, respectively.

Infected cultures were passaged twice a week and uninfected cells added after the appearance of syncytia to amplify the virus. Once the cytopathic effect (CPE) had destroyed > 70% of the cell layer, infected cells and supernatants were treated by three cycles of freezing and thawing (-80°C; +37°C) to enable the release of viral particles. The lysate was cleared by centrifugation (1500 x g for 10 min), filtered through a 0.45 µm pore-size filter, and stored as single-use aliquots at -80°C. Both chimpanzee and gorilla SFV are efficiently detected by GFAB cells [START_REF] Lambert | A new sensitive indicator cell line reveals crosstransactivation of the viral LTR by gorilla and chimpanzee simian foamy viruses[END_REF]. Virus titers were determined by infecting GFAB cells at 30 to 40% confluence in flat-bottom 96-well plates with 30 µl/well of serially-diluted viral solutions prepared in DMEM. After a 2-h incubation, 170 µl DMEM-5% FBS was added to each well. Tests were performed in triplicate. The cells were fixed after 72 h with 0.5% glutaraldehyde in a phosphate-buffered saline solution (PBS) for 10 min at room temperature (RT).

Cells were washed with PBS and incubated 1 h at 37°C with an X-Gal staining solution (2 mM MgCl2; 10 mM Potassium ferricyanide, 10 mM Potassium ferrocyanide; and 0.5 mg/mL 5-Bromo-4-chloro-3-indolyl-B-D-galactopyranoside in PBS). An Ultimate UV Image analyzer (CTL Europe, Bonn, Germany) was used to count X-Gal stained cells. One infectious unit was defined as a blue cell or syncytia.

SFV infections

For microtitration in P96 plates, BHK-21 and GFAB cells were seeded at 5 10 3 cells/P96-well and HT1080 were seeded at 2.5 10 3 cells/P96 well on the day before the infection. Adherent cells were infected at 30 to 40% confluence with 25 µl of undiluted GI-D468 and GII-K74 (5 10 3 infectious unit (IU)/ml). CI-PFV (2 10 6 IU/ml) and CII-SFV7 (3 10 5 IU/ml) stocks were diluted to achieve the same MOI as the gorilla SFV strains in experiments comparing the four strains. The MOIs were calculated as the [inoculum volume x infectious titer]/number of cells seeded per well and were 0.01 for BHK-21 and GFAB cells, and 0.02 for HT1080 cells. We used the same MOI as for microtitration for infections carried out in flasks or six-well plates. Non-adherent cells (10 6 cells) were centrifuged at 500 x g before the addition of the 300 µl of viral stocks to the cell pellet. The MOI were 0.6 for undiluted CI-PFV and 0.0015 for undiluted GI-D468. Infected cultures were passaged twice a week by dilution into fresh culture medium at the same cell density as for the propagation of uninfected cells and maintained up to the occurrence of massive CPE. Or microtitration, GFAB cells were infected in triplicate using the titration protocol and stained after 72 h. Exposure to acidic pH was performed 2 h post-infection by removing the medium, adding PBS at pH 7 or 5.5, and incubation for 1 or 15 min. The PBS was discarded and DMEM with 5% FBS added to the wells. Polyethyleneimine (jetPEI, #101-10, Polyplus), 1,5-dimethyl-1,5-diazaundecamethylene polymethobromide (Polybrene, # TR-1003, Sigma-Aldrich), and diethylaminoethyl dextran (DEAE Dextran, wt 500.000, #D9885 Sigma-Aldrich) were added to the viral inoculum immediately before the infection of GFAB cells. Inhibitors of JAK1/2 (Ruxolitinib, #S1378, Euromedex), TBK-1 (BX795, #S1274, Euromedex), and IKK2 (TPCA-1, #T1452 and BMS345541, #B9935, Sigma-Aldrich) were added to the culture medium 2 h before infection and during the 72h following incubation with the viral inoculum.

Flow cytometry

A murine monoclonal antibody specific for the leader peptide of the SFV envelope (clone P6G11G11 generated by M.L. Linial) was conjugated to Alexa Fluor 647 dye (anti-Env-AF647).

Antibody production, coupling, and purification were performed by RD Biotech, Besançon, France.

Cells were fixed with 2% paraformaldehyde (PFA) diluted in PBS for 10 min at RT, washed in PBS supplemented with 0.1% bovine serum albumin (BSA), permeabilized with 0.5% Triton X-100 in PBS-0.1% BSA for 10 min at RT, washed before addition of anti-Env-AF647 at 20 ng/mL and viability dye (0.5 µl/tube, Live Dead Aqua, #L34957, Life Technologies), and incubated for 30 min at RT. Cells were washed with PBS-0.1% BSA and resuspended in 300 µL PBS-2% PFA. Data were acquired on a Gallios cytometer (Beckman Coulter) and analyzed with Kaluza software. Results are expressed as the percentage among viable cells.

Statistics

We used the Student's t test to compare infection levels between cells treated with acidic or neutral pH, cation polymers, or IFN inhibitors. The number of infected cells/well was determined and the ratio of treated/mock-treated wells calculated. The means and SD from three independent experiments are shown. The t test was used to compare the number of infected cells after polycation exposure versus that after mock treatment.

Figure 1 .

 1 Figure 1. Replication kinetics of SFVs in several cell lines. Infected cultures were split twice a

Figure 2 .

 2 Figure 2. Factors affecting early steps of the replication cycle have no effect on infection

Figure 3 .

 3 Figure 3. Effect of IFN response inhibitors on SFV replication. GFAB cells were treated with

Figure 4 .

 4 Figure 4. Ruxolitinib enhances replication of gorilla SFV but has no effect on CI-PFV. GFAB

Figure 4

 4 Figure 4

Acknowledgments

We thank members of the EPVO research unit for discussions and technical advice. The text has been edited by a native English speaker.

Funding

This work was supported by the Agence Nationale de la Recherche [grant ANR-10-LABX62-IBEID; REEMFOAMY project, ANR 15-CE-15-0008-01]. D.C.F. was personally supported by an Amgen Scholars fellowship. The funding agencies had no role in the study design, generation of results, or writing of the manuscript.

Conflicts of Interest

The authors have no conflicting interests relevant to the study.


Marc Briane
email: mbriane@insa-rennes.fr

Loïc Hervé
email: loic.herve@insa-rennes.fr

Revisiting the asymptotics of the flow for some dynamical systems on the torus

Keywords: dynamical system, flow, torus, Z d -periodic, asymptotics, invariant measure, ergodic theorem, divergence-curl result Mathematics Subject Classification: 37C10, 37C40, 34E10

come

Revisiting the asymptotics of the flow for some dynamical systems on the torus

Introduction

This paper is devoted to the large time asymptotics of the solution X(t, x) to the dynamical system

   ∂X ∂t (t, x) = b(X(t, x)), t ∈ R X(0, x) = x ∈ R d , (1.1)
where b is a C 1 -regular Z d -periodic vector field in R d . More precisely, we focus on the existence of the limit of X(t, x)/t as t → ∞ for x ∈ R d . This problem is strongly connected to the asymptotic behavior of the transport equation with an oscillating velocity

∂u ε ∂t (t, x) + b x ε • ∇ x u ε (t, x) = 0 for (t, x) ∈ [0, ∞) × R d , (1.2)
which is dealt with in [START_REF] Brenier | Remarks on some linear hyperbolic equations with oscillatory coefficients[END_REF][START_REF] Hou | Homogenization of linear transport equations with oscillatory vector fields[END_REF][START_REF] Golse | Moyennisation des champs de vecteurs et EDP[END_REF][START_REF] Golse | On perturbations of dynamical systems and the velocity averaging method for PDEs[END_REF][START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF][START_REF] Briane | Isotropic realizability of fields and reconstruction of invariant measures under positivity properties. Asymptotics of the flow by a nonergodic approach[END_REF]. Otherwise, this question naturally arises in ergodic theory since it involves the flow T t defined by (T t ϕ)(x) := ϕ(X(t, x)) for ϕ continuous and

Z d -periodic in R d , x ∈ R d , (1.3)
which is associated with system (1.1), and the Borel measures µ on the torus Y d := R d /Z d which are invariant for the flow, i.e.

∀ t ∈ R, µ • T t = µ. (1.4)
A strengthened variant of the famous Birkhoff ergodic theorem [5, Theorem 2, Section 1.8] claims that if the flow is uniquely ergodic, i.e. there exists a unique probability measure µ on Y d invariant for the flow, then any continuous Z d -periodic function f satisfies

∀ x ∈ R d , lim t→∞ 1 t t 0 f (X(s, x)) ds = Y d f (y) dµ(y), (1.5)
and the converse actually holds true. In the particular case where f := b, limit (1.5) yields

∀ x ∈ R d , lim t→∞ X(t, x) t = Y d
b(y) dµ(y).

(1.6)

The unique ergodicity condition seems to be a rather restrictive condition on the flows of type (1.1). For example, if the vector field b vanishes at two points x = y in the torus, then any convex combination of the Dirac masses δ x and δ y are invariant for the flow. When b does not vanish in R d , the unique ergodicity of the flow is not clear. This is true in dimension one (see Proposition 3.1) but the situation is more complicated in higher dimension. As best we know the asymptotics of the flow is completely known only in dimension two for a non-vanishing vector field b. Specifically in dimension two, Peirone [START_REF] Peirone | Convergence of solutions of linear transport equations, Ergodic Theory Dynam[END_REF] obtained the asymptotics of the flow through the following alternative:

• Under the existence of a periodic trajectory X(•, y) of (1.1) with respect to the torus (see (1.14)), the limit of (1.6) does exist but may depend on x.

• In the absence of periodic trajectory with respect to the torus, the limit of (1.6) exists independently of x. To this end, Peirone used an one-dimensional ergodicity result transversally to Siegel's curve, but his approach has nothing to do with the unique ergodicity of the flow.

On the other hand, when the two-dimensional flow associated with a non-vanishing vector field b = (b 1 , b 2 ) admits an invariant measure with a positive Z 2 -periodic density σ with respect to the Lebesgue measure, or equivalently, by virtue of Liouville's theorem σb is divergence free in R 2 , then Kolmogorov's theorem [START_REF] Kolmogorov | On dynamical systems with an integral invariant on the torus[END_REF] implies the existence of a diffeomorphism on the torus which transforms the dynamical system (1.1) into the dynamical system

   ∂Y ∂t (t, y) = a(Y (t, y)) ξ, t ∈ R Y (0, x) = y ∈ R 2 , (1.7)
where a is a regular Z 2 -periodic vector field in R 2 and ξ = (ξ 1 , ξ 2 ) is a nonzero constant vector in R 2 . In this two-dimensional setting with the additional assumption that b 1 is nonvanishing, Tassa [START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF] also obtained the asymptotics of the flow through the following alternative which is strongly connected to the above Peirone's alternative:

• If ξ 1 and ξ 2 are rationally dependent, the limit of (1.6) exists but does depend on x.

• If ξ 1 and ξ 2 are rationally independent, or equivalently the so-called rotation number is irrational (see, e.g., [12, Chapter I, Section 4.1]), the limit of (1.6) exists independently of x. The more general result where the vector field b (rather than b 1 ) is nonvanishing with an invariant density σ ∈ C 5 (Y 2 ) is due to Kolmogorov [START_REF] Kolmogorov | On dynamical systems with an integral invariant on the torus[END_REF].

Note that in [START_REF] Peirone | Convergence of solutions of linear transport equations, Ergodic Theory Dynam[END_REF][START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF] the nonvanishing condition of the vector field b is essential to obtain the asymptotics of the flow. Actually, the unique ergodicity of the flow (1.3) associated with b is not needed to get the desired asymptotics (1.6) which is much less restrictive than (1.5). Indeed, our approach consists in replacing the unique ergodicity assumption by the weaker uniqueness condition # Y d b(y) dµ(y) : µ is an invariant probability measure for the flow = 1.

(1.8)

This new condition is obtained by observing that the Birkhoff time average in (1.6) is only addressed (contrary to (1.5)) to the function b. Then, we revisit (see Proposition 2.1) the proof of the existence of an invariant probability measure for the flow defined on a compact space (the torus here). We conclude by proving (see Theorem 2.1) that condition (1.8) turns out to be equivalent to asymptotics (1.6). On the other hand, we show (see Theorem 2.2 and Remark 2.2) that the Liouville theorem satisfied by an invariant measure µ for the flow associated with b can be regarded as a divergence-curl result such that for any

Z d -periodic C 1 -function ψ, Y d b(y) • ∇ψ(y) dµ(y) = 0. (1.9)
Our new approach consists in combining the tools (1.8) and (1.9) to obtain the limit of X(t, x)/t as t → ∞ for some dynamical systems (1.1). So, we get the desired asymptotics for the dynamical system (1.7) (see Proposition 3.2) assuming that the Z d -periodic function a is nonnegative and may vanish contrary to [START_REF] Peirone | Convergence of solutions of linear transport equations, Ergodic Theory Dynam[END_REF][START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF], and that the vector ξ satisfies one of the two following conditions which extend in any dimension the above two-dimensional conditions obtained by Peirone (see Remark 3.3 and Proposition 3.4) and Tassa:

• There exists T > 0 such that T ξ ∈ Z d , which leads us to a limit of X(t, x)/t depending on x.

• For any k ∈ Z d \ {0}, ξ • k = 0, which leads us to the limit

∀ x ∈ R d , lim t→∞ X(t, x) t = a * ξ with a * :=      Y d a -1 (y) dy -1 if a > 0 in R d 0 if a vanishes in R d .
When a vanishes at different points in the torus, the unique ergodicity of the flow (1.5) is not satisfied, while the asymptotics of the flow holds true for b in (1.6). The former asymptotics result easily extends (see Corollary 3.1) to the case where the vector field b is rectifiable to a fixed direction ξ through a diffeomorphism Φ on the torus, i.e. b = a • Φ ∇Φ -1 ξ for some Z d -periodic nonnegative function a. Finally, again using (1.8) and (1.9) we obtain the zero vector limit in (1.6) when b is a current field (see Proposition 3.5), i.e. b = A∇v with A a regular Z d -periodic nonnegative symmetric matrix-valued conductivity and v a regular Z d -periodic potential.

Notation • Y d for d ≥ 1, denotes the d-dimensional torus R d /Z d , which is identified to the cube [0, 1) d in R d . • C k c (R d ) for k ∈ N ∪ {∞}, denotes the space of the real-valued functions in C k (R d ) with compact support. • C k (Y d ) for k ∈ N ∪ {∞}, denotes the space of the real-valued functions f ∈ C k (R d ) which are Z d -periodic, i.e. ∀ k ∈ Z d , ∀ x ∈ R d , f (x + k) = f (x).
• L p (Y d ) for p ≥ 1, denotes the space of the real-valued functions in L p loc (R d ) which are Z d -periodic.

• M (Y d ) denotes the space of the Radon measures on Y d , and M p (Y d ) denotes the space of the probability measures on Y d .

• D (R d ) denotes the space of the distributions on R d .

Definitions and recalls

Let b : R d → R d be a vector-valued function in C 1 (Y d ) d . Consider the dynamical system    ∂X ∂t (t, x) = b(X(t, x)), t ∈ R X(0, x) = x ∈ R d .
(1.10)

The solution X(•, x) of (1.10) which is known to be unique (see, e.g., [START_REF] Hirsch | Differential equations, Dynamical Systems, and an Introduction to Chaos[END_REF]Section 17.4]) is associated with the flow (T t ) t∈R , defined by

T t (ϕ)(x) := ϕ X(t, x) for ϕ ∈ C 0 (Y d ) and x ∈ R d , (1.11)
which satisfies the semi-group property

∀ s, t ∈ R, T s+t = T s • T t , (1.12)
and is well defined in the torus since

∀ t ∈ R, ∀ x ∈ R d , ∀ k ∈ Z d , X(t, x + k) = X(t, x) + k. (1.13)
Property (1.13) follows immediately from the uniqueness of the solution X to (1.10) combined with the Z d -periodicity of b.

A solution X(•, x) to (1.10) is said to be periodic in the torus if there exists τ > 0 and

k ∈ Z d such that ∀ t ∈ R, X(t + τ, x) = X(t, x) + k. ( 1
∀ t ∈ R, ∀ ψ ∈ C 0 (Y d ), Y d T t (ψ)(y) dµ(y) = Y d ψ X(t, y) dµ(y) = Y d ψ(y) dµ(y), (1.15)
or equivalently, the image measure of µ by the flow T t agrees with µ.

2 Some general results

Existence of invariant probability measures for the flow

When a flow preserves the set of the continuous functions on a compact metric space, the existence of an invariant probability measure for the flow is a classical statement which can be derived thanks to a weak compactness argument applied to sequences of probability measures defined from the Birkhoff time averages in (1.5) (see, e.g., [5, Theorem 1, Section 1.8] in the discrete time case). The following result adapts this statement restricting it to the limit points of the Birkhoff time averages for a given function.

Proposition 2.1 Let b ∈ C 1 (Y d ) d .
There exists an invariant probability measure on Y d for the flow T t (1.11) associated with b. Moreover, let g ∈ C 0 (Y d ) and x ∈ R d be fixed, and let (t n ) n∈N ∈ R N be such that lim n t n = ∞. Then, for any limit point a of the sequence

(u n ) n∈N ∈ R N defined by u n := 1 t n tn 0 g(X(s, x)) ds, n ∈ N, (2.1)
there exists a probability measure µ ∈ M p (Y d ) (depending a priori on x and g) which is invariant for the flow T t and satisfies

a = Y d g(y) dµ(y). (2.2)
Proof. As above mentioned the following lemma is classical in ergodic theory. For the reader's convenience its proof is postponed to the Appendix.

Lemma 2.2 Let x ∈ R d , let (r n ) n∈N ∈ R N be such that lim n r n = ∞, and let ν n ∈ M p (Y d ),
n ∈ N, be the probability measure defined by

Y d f (y) dν n (y) = 1 r n rn 0 f (X(s, x)) ds for f ∈ C 0 (Y d ). (2.3)
Then, there exists a subsequence (ν n k ) k∈N of (ν n ) n∈N which converges weakly * to some probability measure µ ∈ M p (Y d ) which is invariant for the flow T t .

Let a be a limit point of the sequence (u n ) n∈N (2.1), namely

a = lim n→∞ 1 t θ(n) t θ(n) 0 g(X(t, x)) dt, for some subsequence (t θ(n) ) n∈N of (t n ) n∈N . Set r n := t θ(n)
, and consider the associated sequence (ν n ) n∈N of probability measures on Y d given by (2.3). By Lemma 2.2 we can extract a subsequence (ν n k ) k∈N which converges weakly * to some invariant measure µ ∈ M p (Y d ) for the flow T t . We thus have

∀ f ∈ C 0 (Y d ), lim k→∞ Y d f (y) dν n k (y) = Y d f (y) dµ(y), which implies in particular that a = lim k→∞ 1 r n k rn k 0 g(X(s, x)) ds = lim k→∞ Y d g(y) dν n k (y) = Y d g(y) dµ(y).
Remark 2.1 The second assertion of Proposition 2.1 may provide a simple way to find the limit of Birkhoff 's time averages for a specific function, when the general form of the limit in Birkhoff 's theorem is unknown. For instance, if the support of any invariant probability measure for the flow is contained in a subset

F of Y d and if g ∈ C 0 (Y d ) vanishes on F , then ∀ x ∈ Y d , lim t → ∞ 1 t t 0
g(X(s, x)) ds = 0, since any limit point of these averages is zero by (2.2).

A criterium for the asymptotics of the flow

It is known (see, e.g., [5, Theorem 2, Section 1.8] for the discrete case) that the uniqueness of an invariant measure µ ∈ M p (Y d ) for the flow T t is equivalent to the pointwise property:

∀ x ∈ R d , ∀ f ∈ C 0 (Y d ), lim t→∞ 1 t t 0 f (X(s, x)) ds = Y d f (y) dµ(y).
(2.4)

The following result which is new as the best we know, allows us to restrict condition (2.4) to f = b and to derive the asymptotics of the flow at each point in R d , by assuming the uniqueness of the averages of b with respect to the invariant probability measures on Y d for the flow rather than the uniqueness of an invariant measure for the flow.

C b = {ζ} ⇔ ∀ x ∈ R d , lim t→∞ X(t, x) t = ζ. ( 2
C b is a closed set in R d . We also have C b ⊂ [0, b ∞ ], so that C b is a compact set of R d . Now, assume that C b = {ζ}. Let x ∈ R d , let (t n ) n∈N ∈ R N be such that lim n t n = ∞,
∀ x ∈ R d , lim t→∞ X(t, x) t = lim t→∞ 1 t t 0 b(X(s, x)) ds = ζ.
Conversely, assume that the right-hand side of (2.6) holds, which implies that

∀ x ∈ R d , lim t→∞ 1 t t 0 b(X(s, x)) ds = ζ.
Then, integrating over Y d the former equality with respect to any probability measure µ ∈ I b , then applying successively Lebesgue's dominated convergence theorem and Fubini's theorem, we get that

ζ = lim t→∞ Y d 1 t t 0 b(X(s, x)) ds dµ(x) = lim t→∞ 1 t t 0 Y d b(X(s, x)) dµ(x) ds = Y d b(x) dµ(x).
which shows that C b = {ζ}. This concludes the proof of (2.6).

Liouville's theorem and a divergence-curl result

Liouville's theorem provides a criterium for a probability measure on a smooth compact manifold in R d (see, e.g., [5, Theorem 1, Section 2.2]) to be invariant for the flow. The next result revisits this theorem in M p (Y d ) in association with a divergence-curl result on the torus.

Theorem 2.2 Let b ∈ C 1 (Y d ) d and let µ ∈ M p (Y d ). We define the Radon measure μ ∈ M (R d ) on R d by R d ϕ(x) dμ(x) := Y d ϕ (y) dµ(y) where ϕ (•) := k∈Z d ϕ(• + k) for ϕ ∈ C 0 c (R d ). (2.7)
Then, µ is invariant for the flow T t if, and only if, one of the two following conditions is satisfied:

div (μ b) = 0 in D (R d ), (2.8)
∀ ψ ∈ C 1 (Y d ), Y d b(y) • ∇ψ(y) dµ(y) = 0.
(2.9)

Proof. Assume that µ is invariant for the flow, i.e. (1.15). Let ϕ ∈ C 1 c (R d ). Since by (1.13) we have for any t ∈ R and y

∈ R d , ϕ(X(t, •)) (y) = k∈Z d ϕ(X(t, y + k)) = k∈Z d ϕ(X(t, y) + k) = ϕ (X(t, y)), (2.10)
it follows from (2.7) and the invariance of µ that

∀ t ∈ R, R d ϕ(X(t, x)) dμ(x) = Y d ϕ(X(t, •)) (y) dµ(y) = Y d ϕ (X(t, y)) dµ(y) = Y d ϕ (y) dµ(y) = R d ϕ(x) dμ(x).
Taking the derivative of the former expression with respect to t, we get that

∀ t ∈ R, R d b(X(t, x)) • ∇ϕ(X(t, x)) dμ(x) = 0, which at t = 0 yields ∀ ϕ ∈ C 1 c (R d ), R d b(x) • ∇ϕ(x) dμ(x) = 0, (2.11)
namely the variational formulation of the distributional equation (2.8).

Conversely, assume that equation (2.8) holds true. Let ϕ ∈ C 1 c (R d ) and define the function φ ∈ C 1 (R × R d ) by φ(t, x) := ϕ(X(t, x)). By the semi-group property (1.12) we have for any

s, t ∈ R and x ∈ R d , ∂ ∂s φ(s + t, X(-s, x)) = ∂ ∂s φ(t, x) = 0 = ∂φ ∂s (s + t, X(-s, x)) -b(X(-s, x)) • ∇ x φ(s + t, X(-s, x)),
which at s = 0 gives the classical transport equation

∀ t ∈ R, ∀ x ∈ R d , ∂φ ∂t (t, x) = b(x) • ∇ x φ(t, x). (2.12) Hence, since ϕ(X(t, •)) is in C 1 (R d
) and has a compact support independent of t when t lies in a compact set of R, we deduce from (2.12) and (2.8) that

∀ t ∈ R, d dt R d ϕ(X(t, x)) dμ(x) = R d b(x) • ∇ x ϕ(X(t, x)) dμ(x) = 0,
or equivalently,

∀ t ∈ R, R d ϕ(X(t, x)) dμ(x) = R d ϕ(x) dμ(x).
On the other hand, we have the following result. Hence, using relation (2.10) and definition (2.7) we get that for any ψ ∈ C ∞ (Y ),

∀ t ∈ R, Y d ψ(X(t, y)) dµ(y) = Y d ϕ (X(t, y)) dµ(y) = R d ϕ(X(t, x)) dμ(x) = R d ϕ(x) dμ(x) = Y d ψ(y) dµ(y),
which shows that µ is invariant for the flow. We have just proved the equivalence between the invariance of µ for the flow and the distributional equation (2.8) satisfied by μ. Finally, the equivalence between (2.8), or equivalently (2.11), and (2.9) is a straightforward consequence of the relation Remark 2.2 Equation (2.9) can be considered as the divergence free of the vector-valued measure µ b in the torus Y d , while equation (2.8) is exactly the divergence free of the vector-valued measure μ b in the space R d . Equation (2.9) is also equivalent to

∀ ϕ ∈ C 1 c (R d ), R d b(x) • ∇ϕ(x) dμ(x) =
∀ ∇ψ ∈ C 0 (Y d ) d , Y d b(y) • ∇ψ(y) dµ(y) = Y d b(y) dµ(y) • Y d ∇ψ(y) dy , (2.13) since ∇ψ ∈ C 0 (Y d ) d ⇔ x → ψ(x) -x • Y d ∇ψ(y) dy ∈ C 1 (Y d ).
Condition (2.13) has to be regarded as a divergence-curl result combining the invariant measure µ for the divergence free vector field µ b and the Lebesgue measure for the gradient field ∇ψ.

Application to the asymptotics of the flow

First of all, we apply the tools of Section 2 to the one-dimensional case.

The one-dimensional case

We have the following result.

Proposition 3.1 Let b ∈ C 1 (Y 1 )
. We have the following alternative:

(i) If b = 0 sur Y 1 ,
∀ x ∈ R, lim t→∞ X(t, x) t = 0. (3.2)
Proof. Case (i). Let µ ∈ M p (Y 1 ) be an invariant measure for the flow T t . By the condition (2.8) of Theorem 2.2 there exists a constant C ∈ R such that μ b = C in R, or equivalently,

∀ ϕ ∈ C 0 c (R), R ϕ(x) b(x) dμ(x) = C R ϕ(x) dx.
Hence, by the definition (2.7) of μ we have

∀ ϕ ∈ C 0 c (R), Y 1 ϕ (y) b(y) dµ(y) = R ϕ(x) b(x) dμ(x) = C R ϕ(x) dx = C Y 1 ϕ (y) dy.
Therefore, from Lemma 2.3 we deduce that As a consequence the set I b defined by (2.5) is a singleton, and C b = {b}. Therefore, due to the equivalence (2.6) of Theorem 2.1 we obtain the desired asymptotics (3.1). We could also have concluded directly thanks to the unique ergodicity theorem. However, the unique ergodicity theorem does not apply in the following case, while Theorem 2.1 does.

∀ ψ ∈ C 0 (Y 1 ),
Case (ii). Let µ ∈ M p (Y 1 ) be an invariant measure for the flow T t . The equality μ b = C still holds true in R for some constant C ∈ R, as well as equality (3.3). Take the function ψ := (|b| + ε) -1 for ε > 0, in equality (3.3), and make ε tend to 0. Then, using that b is regular and vanishes in Y 1 (which implies that 1/b / ∈ L 1 (Y d )), and applying successively Beppo-Levi's theorem and (3.3), it follows that As a by-product, using Lebesgue's dominated convergence theorem in the third integral of (3.4), we also get the equality

|C| × ∞ = |C| × Y 1 dy |b(y)| = lim ε→0 Y 1 C dy |b(y)| + ε = lim inf
(i) If ξ • k = 0 for any k ∈ Z d \ {0}, then we have ∀ x ∈ R d , lim t→∞ X(t, x) t = a * ξ with a * := a if a > 0 in Y d 0 if a vanishes in Y d , (3.7)
where a is the harmonic mean of a.

(ii) If there exists T > 0 such that T ξ ∈ Z d , then we have

∀ x ∈ R d , lim t→∞ X(t, x) t = a * (x) ξ with a * (x) :=        1 T T 0 ds a sξ + Π ξ ⊥ (x) -1 ξ if ∀ u ∈ R, a uξ + Π ξ ⊥ (x) = 0 0 if ∃ u ∈ R, a uξ + Π ξ ⊥ (x) = 0, (3.8)
where Π ξ ⊥ denotes the projection on the hyperplane ξ ⊥ orthogonal to ξ.

Remark 3.1 The two-dimensional framework is completely covered by the disjoint cases (i) and (ii) of Proposition 3.2, since

∃ k ∈ Z d \ {0}, ξ • k = 0 ⇔ ∃ T > 0, T ξ ∈ Z 2 .
This equivalence does not hold in higher dimension.

Proof of Proposition 3.2. Case (i). Assume that for any

k ∈ Z d \ {0}, ξ • k = 0.
First assume that a does not vanish in Y d . Let x ∈ R d . Since the vector field b is parallel to the fixed direction ξ, we have

∀ t ∈ R, X(t, x) = X(t, x) • ξ ξ + Π ξ ⊥ X(t, x) and Π ξ ⊥ X(t, x) = Π ξ ⊥ (x), (3.9) which implies that    ∂(X • ξ) ∂t (t, x) = a (X • ξ)(t, x) ξ + Π ξ ⊥ (x) , t ∈ R (X • ξ)(0, x) = x • ξ. (3.10)
Then, the solution X • ξ to equation (3.10) is given by

∀ t ∈ R, X(t, x) • ξ = F -1 x t + F x (x • ξ) where F x (t) := t 0 ds a s ξ + Π ξ ⊥ (x) . (3.11)
By approximating in C 0 (Y d ) the continuous Z d -periodic function 1/a by Fejér's type trigonometric polynomials, and noting that by hypothesis

∀ k ∈ Z d \ {0}, lim t→∞ 1 t t 0 e -2iπ s ξ•k ds = 0, it follows that lim t→∞ F x (t) t = Y d dy a(y) = 1 a , which taking into account that F x X(t, x) • ξ ∼ t→∞ t with X(t, x) • ξ → ∞ as t → ∞, implies that lim t → ∞ X(t, x) • ξ t = a.
Therefore, we get that

∀ x ∈ R d , lim t → ∞ X(t, x) t = lim t → ∞ X(t, x) • ξ ξ + Π ξ ⊥ (x) t = a ξ.
Otherwise, if a vanishes in Y d let us prove that

∀ µ ∈ I aξ , Y d a(y) dµ(y) = 0,
or equivalently, C aξ = {0}. Then, Theorem 2.1 will allow us to conclude. Assume by contradiction that there exists a measure µ ∈ I aξ such that

Y d a(y) dµ(y) > 0.
Set a ε := a + ε for ε > 0, and define the probability measure (recall that a ≥ 0)

dµ ε (x) := Y d a(y) a ε (y) dµ(y) -1 a(x) a ε (x) dµ(x), (3.12)
which is well defined since

Y d a(y) a ε (y) dµ(y) ≥ 1 a ∞ + ε Y d a(y) dµ(y) > 0.
Since µ is invariant for the flow associated with a ξ, by equality (2.9) we have

∀ ϕ ∈ C 1 (Y d ), Y d a ε (x) ξ • ∇ϕ(x) dµ ε (x) = Y d a(y) a ε (y) dµ(y) -1 Y d a(x) ξ • ∇ϕ(x) dµ(x) = 0,
hence µ ε ∈ I aεξ . But from the former case a > 0 combined with Theorem 2.1 we deduce that

C aεξ = Y d a ε (y) µ ε (dy) ξ = {a ε ξ} or equivalently Y d a ε (y) µ ε (dy) = a ε .
This combined with the expression of a ε (x) dµ ε (x) given by (3.12) leads us to the equality

Y d a(y) a ε (y) dµ(y) a ε = Y d a(y) a ε (y) dµ(y) Y d a ε (x) µ ε (dx) = Y d a(x) µ(dx).
Then, applying Beppo-Levi's theorem and using that a is regular and vanishes in

Y d (which implies that 1/a / ∈ L 1 (Y d )), it follows that 0 < Y d a(x) µ(dx) = Y d a(y) a ε (y) dµ(y) a ε ≤ a ε -→ ε→0 Y d dy a (y) -1
= 0, which yields a contradiction. Therefore, we get that C aξ = {0}, and by the convergence (2.6) of Theorem 2.1 we obtain that

∀ x ∈ R d , lim t → ∞ X(t, x) t = 0.
Case (ii). Now, assume that there exists

T > 0 such that T ξ ∈ Z d . Let x ∈ R d . We have to distinguish two cases. If a uξ + Π ξ ⊥ (x) > 0 for any u ∈ R, then by the T -periodicity of u → a uξ + Π ξ ⊥ (x) we have lim t→∞ F x (t) t = 1 T T 0 ds a sξ + Π ξ ⊥ (x) , hence lim t → ∞ X(t, x) t = 1 T T 0 ds a sξ + Π ξ ⊥ (x) -1 ξ.
On the contrary, if there exists u 0 ∈ R such that a u 0 ξ + Π ξ ⊥ (x) = 0, then the part (ii) of Proposition 3.1 applies to the one-dimensional solution X •ξ to (3.10), where u → a uξ+Π ξ ⊥ (x) is T -periodic and vanishes at the point u 0 . We thus deduce that lim t→∞ X(t, x) • ξ t = 0, then lim t→∞ X(t, x) t = 0, which concludes the proof.

Rectification to a fixed direction

In this section we extend the result of the former section thanks to a diffeomorphism on the torus.

Definition 3.3 A mapping Φ ∈ C 1 (R d ) d
is said to be a C 1 -diffeomorphism on the torus if Φ satisfies the following conditions:

• det(Φ(x)) = 0 for any x ∈ R d ,
• there exist a matrix A ∈ Z d×d with | det(A)| = 1, and a

Z d -periodic mapping Φ ∈ C 1 (Y d ) d such that ∀ x ∈ R d , Φ(x) = Ax + Φ (x). (3.13)
Note that the invertibility of A and the periodicity of Φ in (3.13) imply that Φ is a proper function (i.e., the inverse image by the function of any compact set in R d is a compact set). Hence, by virtue of Hadamard-Caccioppoli's theorem [START_REF] Caccioppoli | Sugli elementi uniti delle trasformazioni funzionali: un teorema di esistenza e unicit ed alcune sue applicazioni[END_REF] (also called Hadamard-Lévy's theorem) the mapping Φ is actually a C 1 -diffeomorphism on R d . Also noting that due to A -1 ∈ Z d×d , we have

∀ k ∈ Z d , ∀ x ∈ R d , Φ(x + k) -Φ(x) = Ak ∈ Z d Φ -1 (x + k) -Φ -1 (x) = A -1 k ∈ Z d ,
Φ well defines an isomorphism on the torus. Using a diffeomorphism on the torus the result of Proposition 3.2 can be extended to the following general result. Then, if ξ satisfies one of the two conditions of Proposition 3.2 we get that

∀ x ∈ R d , lim t→∞ X(t, x) t = a * (Φ(x)) A -1 ξ, (3.15)
where a * (y) is given by formula (3.7) or formula (3.8). (3.17)

By the chain rule and (3.14) we have

   ∂Y ∂t (t, y) = ∇Φ(X(t, x)) ∂X ∂t (t, x) = ∇Φ(X(t, x)) b(X(t, x)) = a(Φ(X(t, x))) ξ = a(Y (t, y)) ξ Y (0, y) = Φ(X(0, x)) = Φ(x) = y,
which shows that Y is the solution to the dynamical system associated with the vector field a ξ. Hence, if ξ satisfies one of the two conditions of Proposition 3.2, we get the asymptotics of the flow

∀ y ∈ R d , lim t→∞ Y (t, y) t = a * (y) ξ,
where a * (y) is given by formula (3.7) or formula (3.8). Therefore, by the boundedness of Φ in (3.13) we have for any x ∈ R d and y := Φ(x),

X(t, x) t = Φ -1 (Y (t, y)) t = A -1 A Φ -1 (Y (t, y)) t = A -1 Y (t, y) t + O(1) t -→ t→∞ a * (y) A -1 ξ,
which yields the desired asymptotics (3.15).

Remark 3.3 Peirone [START_REF] Peirone | Convergence of solutions of linear transport equations, Ergodic Theory Dynam[END_REF]Theorem 3.1] proved that in dimension two and for a nonvanishing vector field b ∈ C 1 (Y 2 ) 2 , there is no periodic solution in R 2 to system (1.10) (see the definition after (1.14)), and that the limit of X(t, x)/t as t → ∞ does exist for any x ∈ R 2 with the following alternative:

• If system (1.10) has no periodic solution in the torus according to (1.14), then the limit of X(t, x)/t as t → ∞ is independent of x ∈ R 2 as in (3.7).

• If system (1.10) has a periodic solution in the torus, then the limit of X(t, x)/t as t → ∞ may depend on x ∈ R 2 as in (3.8).

Actually, in any dimension the alternative on the vector ξ in Corollary 3.1 (see also Proposition 3.2) is naturally connected to the former alternative due to the following result.

Proposition 3.4 In the framework of Corollary 3.1 assume in addition that

∃ x ∈ R d , ∀ t ∈ R, a t ξ + Π ξ ⊥ (x) > 0. (3.18)
Then, the following equivalence holds: Proof. First note that the definition (3.13) of the C 1 -diffeomorphism on the torus Φ implies that a solution X(•, x) is periodic in the torus if, and only if, the function Y (•, Φ(x)) is periodic in the torus. Therefore, it is enough to prove the result in the case b = a ξ.

∃ T > 0, T ξ ∈ Z d ⇔
If system (1.10) with b = a ξ has a periodic solution X(•, x) in the torus but not in R d , then there exists τ > 0 and k ∈ Z d \ {0} such that by (3.9) we have for any t ∈ R,

Π ξ ⊥ (x) = Π ξ ⊥ (X(t + τ, x)) = Π ξ ⊥ (X(t, x) + k) = Π ξ ⊥ (X(t, x)) + Π ξ ⊥ (k) = Π ξ ⊥ (x) + Π ξ ⊥ (k).
Hence, we obtain that Π ξ ⊥ (k) = 0, or equivalently, there exists T > 0 such that T ξ = ±k ∈ Z d .

Conversely, assume that (3.18) is satisfied for some x ∈ R d , and there exists T > 0 such that T ξ = k ∈ Z d . The solution X(•, x) of (1.10) is given by (3.9) and (3.11). Define

τ := T 0 ds a s ξ + Π ξ ⊥ (x)
.

Then, by the T -periodicity of the function t → a t ξ + Π ξ ⊥ (x) we have

F x (x • ξ + T ) = x•ξ+T 0 ds a s ξ + Π ξ ⊥ (x) = x•ξ 0 ds a s ξ + Π ξ ⊥ (x) + x•ξ+T x•ξ ds a s ξ + Π ξ ⊥ (x) = F x (x • ξ) + τ = F x X(τ, x) • ξ by (3.11), which implies that X(τ, x) • ξ = x • ξ + T . Moreover, the functions t → X(t + τ, x) • ξ and t → X(t, x) • ξ + T are solutions to the system z (t) = a z(t) ξ + Π ξ ⊥ (x)
, and agree at t = 0. Hence, by a uniqueness argument these two solutions are equal, which yields

∀ t ∈ R, X(t + τ, x) • ξ = X(t, x) • ξ + T = X(t, x) + k • ξ.
Moreover, again by (3.9) and recalling that Π ξ ⊥ (k) = 0 we have

∀ t ∈ R, Π ξ ⊥ X(t + τ, x) = Π ξ ⊥ (x) = Π ξ ⊥ X(t, x) = Π ξ ⊥ X(t, x) + k .
The two previous identities imply that

∀ t ∈ R, X(t + τ, x) = X(t, x) + k.
namely, X(•, x) is periodic in the torus.

Consider the vector field b(x) := a(x)e 1 with a(x) := 1 π cos 2 (πx 1 ) ≥ 0 for x ∈ R d . Here, the vector ξ = e 1 clearly satisfies the left-hand side of (3.19). However, since ∀ x ∈ R d , a 1 2 e 1 + Π e ⊥ 1 (x) = 1 π cos 2 ( π 2 ) = 0, condition (3.18) is not satisfied. Moreover, for any x ∈ R d the solution X(•, x) to (1.10) is given by

       X 1 (t, x) = x 1 if x 1 = 1 2 + n, n ∈ Z 1 π arctan(t + tan(πx 1 )) + n if x 1 ∈ (-1 2 + n, 1 2 + n), n ∈ Z X i (t, x) = x i for i ≥ 2 (t, x) ∈ R×R d .
It follows that any periodic solution in the torus is necessarily stationary and a fortiori periodic in R d in the sense of (1.14). Therefore, the implication (⇒) of (3.19) does not hold.

The case of a current field

The following result deals with the case of a current field b = A∇v defined with some matrixvalued conductivity A and some electric field ∇v with zero average. Hence, we get that

∀ t ∈ R, ∀ f ∈ C 0 (Y d ), Y d (T t f )(y) dµ(y) = Y d f (y) dµ(y),
which implies that µ is invariant for the flow T t .

Theorem 2 . 1

 21 Let b ∈ C 1 (Y d ) d . Define the sets I b := µ ∈ M p (Y d ) : µ is invariant for the flow T t and C b := Y d b dµ : µ ∈ I b . (2.5) Then, I b is a nonempty set, and C b is a nonempty compact convex set of R d . Moreover, the following equivalence holds for any ζ ∈ R d ,

. 6 )

 6 Proof. By virtue of Proposition 2.1 the sets I b and C b are nonempty. The set C b is a convex set in R d since I b is clearly convex. Using the compactness of M p (Y d ) for the weak * topology, we get that

  and define the sequence (u n ) n∈N by (2.1) with the function g := b • ξ for ξ ∈ R d . Let a be a limit point of the sequence (u n ) n∈N . By Proposition 2.1 there exists an invariant measure µ ∈ M p (Y d ) for the flow T t satisfying a = Y d b(y) • ξ dµ(y), which by hypothesis implies that a = ζ • ξ. Hence, ζ • ξ is the unique limit point of the bounded sequence (u n ) n∈N which thus converges to ζ • ξ. Therefore, due to the arbitrariness of the sequence (t n ) n∈N we obtain that

Lemma 2 . 3 ([ 2 ], Lemma 3 . 5 )

 23235 For any smooth Z d -periodic function ψ ∈ C ∞ (Y d ), there exists a smooth function with compact support ϕ ∈ C ∞ c (R d ) such that ψ = ϕ .

  Y d b(y) • ∇ϕ (y) dµ(y) which itself follows from [b • ∇ϕ] = b • ∇ϕ and (2.7), combined with Lemma 2.3.

  then b/b(y) dy is the unique invariant measure for the flow T t associated with b, and ∀ x ∈ R, lim t→∞ If b does vanish in Y 1 , then

Y 1 ψ 1 ψ

 11 (y) b(y) dµ(y) = C Y (y) dy. (3.3) Taking ψ = 1/b in (3.3) we get that C = b, which implies that µ(dy) = b/b(y) dy is thus the unique invariant measure for the flow.

ε→0 Y 1 b

 1 (y) |b(y)| + ε dµ(y) ≤ 1, (3.4) which implies that C = 0. Finally, taking ψ = 1 in the equality (3.3) with C = 0, leads us to C b = {0}, which by virtue of the equivalence (2.6) of Theorem 2.1 yields the asymptotics of the flow (3.2).

1 Proposition 3 . 2

 132 The case with a fixed direction This section deals with the case where the vector field b has a fixed direction, namely b(y) = a(y) ξ, y ∈ Y d , (3.6) for some nonnegative function a ∈ C 1 (Y d ) which may vanish, and for a given vector ξ ∈ R d with |ξ| = 1. We have the following result. Let b ∈ C 1 (Y d ) d be given by (3.6) with a ≥ 0 and ξ ∈ R d with |ξ| = 1.

Corollary 3 . 1

 31 Consider a vector field b such that there exist a C 1 -diffeomorphism Φ on the torus given by (3.13), a nonnegative function a ∈ C 1 (Y d ) d and a vector ξ ∈ R d with |ξ| = 1, such that ∀ x ∈ R d , b(x) = a(Φ(x)) ∇Φ(x) -1 ξ. (3.14)

Remark 3 . 2 A

 32 similar result as Corollary 3.1 was proved by Tassa[START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF] Theorem 2.1] in dimension two assuming that the first coordinate b 1 of b does not vanish in R 2 , and that there exists an invariant measure for the flow associated with b, having a density in C 1 (Y 2 ) with respect to Lebesgue's mesure. More precisely, under these two conditions Tassa showed in the spirit of Kolmogorov's theorem the existence of a C 1 -diffeomorphism on the torus satisfying (3.14) with the matrix A = I 2 . In the present case, we only assume the weaker rectification formula (3.14) in any dimension d ≥ 1, with any diffeomorphism Φ on the torus and a possibly vanishing nonnegative function a. Conversely, consider a vector field b ∈ C 1 (Y d ) d satisfying (3.14) with a > 0. By making the change of variables ϕ(x) = ψ(y) with y = Φ(x) for any function ϕ ∈ C ∞ c (R d ), it is easy to check that R d det(∇Φ(x)) a(Φ(x)) b(x) • ∇ϕ(x) dx = R d ξ • ∇ψ(y) dy = 0, which implies that div (σb) = 0 in R d with σ := Since the function σ is Z d -periodic, by virtue of Theorem 2.2 (3.16) means that the density probability measure σ(x) dx is invariant for the flow associated with b. Therefore, the rectification formula (3.14) of a nonvanishing vector field b can be regarded as a (more restrictive) substitute to (3.16) in dimension d > 2 for which Kolmogorov's theorem does not apply. Proof of Corollary 3.1. Define Y (t, y) := Φ(X(t, x)) for (t, x) ∈ R × R d with y := Φ(x).

Proposition 3 . 5 r n k rn k 0 fτ k τ k 0 ffτ k t 0 f

 35000 Assume that b = A∇v in Y d with A ∈ C 1 (Y d ) d×d , A = A T ≥ 0 in Y d , v ∈ C 2 (Y d ) d . (3.20)Then, the flow T t associated with b satisfies the asymptotics∀ x ∈ R d , lim t→∞ X(t, x) t = 0. Proof. Let µ ∈ M p (Y d )be an invariant measure for the flow T t . By the div-curl relation (2.13) combined with the zero average of ∇v, we have Y d A(y)∇v(y) • ∇v(y) ≥0 µ(dy) = Y d b(y) • ∇v(y) µ(dy) = 0, which implies that A∇v • ∇v = 0 µ-a.e. in Y d . Since the matrix-valued A is symmetric and nonnegative, by the Cauchy-Schwarz inequality we deduce that A∇v = 0 µ-a.e. in Y d , and thus Y d b(y) µ(dy) = Y d A(y)∇v(y) µ(dy) = 0. Therefore, C b = {0} and the equivalence (2.6) of Theorem 2.1 allows us to conclude.A Proof of Lemma 2.2Since Y d is a compact metrizable space, there exists a subsequence (ν n k ) k∈N of (ν n ) n∈N which converges weakly * to some probability measure µ ∈ M p (Y d ), namely for anyf ∈ C 0 (Y d ), Y d f (y) dν n k (y) = 1 (X(s, x)) ds -→ k→∞ Y d f (y) dµ(y). (A.1)Let us prove that µ is invariant for the flow T t . For the sake of simplicity denote τ k := r n k andµ k := ν n k . Let t ∈ R and f ∈ C 0 (Y d ).By the semi-group property of the flow (1.12) we haveY d (T t f )(y) dµ k (y) = 1 (X(s + t, x)) ds.By the change of variable r = s + t, it follows thatY d (T t f )(y) dµ k (y(X(r, x)) dr + 1 τ k t+τ k τ k f (X(r, x)) dr -1 (X(r, x)) drSince f is bounded and t ∈ R is fixed, we deduce from (A.1) that lim k→∞ Y d (T t f )(y) dµ k (y) = Y d f (y) dµ(y). However, by the definition of µ we also have lim k→∞ Y d (T t f )(y) dµ k (y) = Y d (T t f )(y) dµ(y).

  .14) If k = 0 the solution is said to be periodic in R d . A measure µ in M p (Y d ) is said to be invariant for the flow T t (see, e.g., [5, Chap. 2]) if


Muamer Kadic
email: muamer.kadic@gmail.com

Andre Diatta

Tobias Frenzel

Sebastien Guenneau

Martin Wegener

André Diatta

Static chiral Willis continuum mechanics for three-dimensional chiral mechanical metamaterials

I. INTRODUCTION

In one dimension, the scalar spring constant in Hooke's law connects forces and displacements. As a generalization towards three-dimensional continuum mechanics [START_REF] Banerjee | An Introduction to Metamaterials and Waves in Composites[END_REF], the rank-4 linear Cauchy elasticity tensor C connects the rank-2 stress tensor σ and the rank-2 strain tensor . In general, linear Cauchy elasticity comprises up to 21 independent nonzero parameters describing possible linear deformations of elastic bodies in three dimensions [START_REF] Sommerfeld | Mechanics of Deformable Bodies[END_REF][START_REF] Authier | International Tables for Crystallography Volume D: Physical Properties of Crystals[END_REF][START_REF] Milton | The Theory of Composites[END_REF][START_REF] Milton | Extending the Theory of Composites to Other Areas of Science[END_REF][START_REF] Laude | Phononic Crystals: Artificial Crystals for Sonic, Acoustic, and Elastic Waves[END_REF]. For cubic crystals, which are characterized by four threefold rotational axes, only three parameters remain [START_REF] Authier | International Tables for Crystallography Volume D: Physical Properties of Crystals[END_REF].

However, linear Cauchy elasticity essentially only grasps the displacements u(r) of infinitesimally small volume elements (of "points," associated with position vectors r) within a fictitious continuum. Linear Cauchy elasticity therefore misses certain degrees of freedom in artificial threedimensional periodic microlattices or metamaterials, for which the unit cell has a finite extent rather than being approximately pointlike such as atoms in an ordinary crystal of macroscopic size [7,8]. Such missed degrees of freedom have recently become particularly obvious in chiral threedimensionally periodic mechanical metamaterial structures [see Fig. 1(a)] for which linear Cauchy elasticity fails to describe any effect of chirality, whereas prominent twist effects have been observed experimentally in the static case [7]. In contrast, micropolar Eringen elasticity [see Fig. 1(b)] has been able to describe these experimental findings [7] as well as others for achiral media [8]. Cosserat elasticity [START_REF] Eringen | Elastodynamics[END_REF] can be seen as a special case of Eringen micropolar elasticity.

Cosserat elasticity and micropolar Eringen elasticity [START_REF] Eringen | Elastodynamics[END_REF][START_REF] Bigoni | [END_REF], micromorphic Eringen elasticity [START_REF] Eringen | Elastodynamics[END_REF]11], strain-gradient theories [12][13][START_REF] Marrec | A continuous model for the wave scattering by a bounded defective domain[END_REF], and yet more advanced approaches [START_REF] Seppecher | [END_REF] are not the only possible generalizations of linear Cauchy elasticity though. It is therefore interesting and relevant to ask which generalizations other than Eringen's can describe the effects of chirality observed in recent experiments [7].

In this paper, in Sec. II, we start with the static version of a generalization of linear Cauchy elasticity following Willis [16,[START_REF] Willis | Elasticity theory of composites[END_REF]. Aiming at describing recent experiments [7], we focus on the case of three-dimensional homogeneous cubic crystals without centrosymmetry, in which case the terms beyond linear Cauchy elasticity can be parameterized by a single scalar parameter. In Sec. III, we discuss numerical solutions. We find that the resulting behavior qualitatively describes the push-to-twist conversion effects observed in recent experiments [see Fig. 1(c)] and that it can be connected to a characteristic length scale. We conclude in Sec. IV.

II. GENERALIZED STATIC LINEAR CAUCHY ELASTICITY

In the static case, all forces must balance. For simplicity, we omit external forces in all formulas throughout this paper. Hence, the divergence of the stress tensor σ = C : is zero, ∇ • σ = 0. The colon symbol : denotes a double contraction; the dot • denotes a contraction between two tensors. Linear Cauchy elasticity reduces to the compact equation

∇ • (C : ) = 0, ( 1
)
where C is the rank-4 elasticity tensor, with components C i jkl (i, j, k, l = 1, 2, 3) in Cartesian coordinates and SI units of Pa, and is the dimensionless symmetric rank-2 strain tensor with components i j = ji [START_REF] Sommerfeld | Mechanics of Deformable Bodies[END_REF]. The strain tensor can be connected to the gradient of the displacement vector field u = u(r) with components

u i (i = 1, 2, 3) via [2] = 1 2 [∇u + (∇u) T ], (2)
FIG. 1. Cuboid beams with volume L × L × 2L, with side length L = 500 μm, are subject to uniaxial loading along the negative z direction. In linear Cauchy elasticity, the beam compresses and can expand or contract laterally, depending on its Poisson ratio (not depicted). However, a twist is forbidden in linear Cauchy elasticity, even if the underlying crystal symmetry would allow for a twist. (a) Finite-element calculation for a chiral metamaterial microstructure exhibiting a twist behavior [7]. (b) Same as (a), but calculated using chiral micropolar Eringen elasticity [7]. where the superscript T refers to the transposed quantity. The linear Cauchy elasticity tensor obeys the minor symmetries (C i jkl = C jikl = C i jlk ) and the major symmetries (C i jkl = C kli j ) [START_REF] Sommerfeld | Mechanics of Deformable Bodies[END_REF]. As a result, the strain tensor in (1) can equivalently be replaced by the gradient of the displacement vector ∇u, i.e., ∇ • (C : ∇u ) = 0 .

(

Linear Cauchy elasticity does not describe effects of chirality at all [START_REF] Eringen | Elastodynamics[END_REF]. This fact can immediately be seen by recalling that all even-rank tensors (such as the rank-2 stress tensor, the rank-4 elasticity stress tensor, and the rank-2 strain tensor) are invariant under space inversion operations, r → -r [START_REF] Eringen | Elastodynamics[END_REF]. Thus, (3) does not change under space inversion, which brings one from a left-handed to a right-handed medium (or vice versa).

Here, we consider Willis elasticity [16,[START_REF] Willis | Elasticity theory of composites[END_REF] as a candidate for describing chiral effects. Willis elasticity has been around for many years [16,[START_REF] Willis | Elasticity theory of composites[END_REF]. The Willis continuum-mechanics equations have mostly been used in the context of dynamic effects (or elastic waves) [16][START_REF] Willis | Elasticity theory of composites[END_REF][START_REF] Milton | [END_REF][19][20][21][22][23][24][25][26][27][28]. We will use them in the static regime. We mention in passing that Milton, Briane, and Willis have proven mathematically in 2006 that the Willis equations are form invariant under general curvilinear spatial coordinate transformations [29,30]. However, their discussion concerning the form invariance of linear Cauchy elasticity has raised some controversy in the literature [31][32][33]. Completely independent of this controversy, linear Willis elasticity remains an interesting generalization of linear Cauchy elasticity. In the static limit, i.e., for angular frequency ω = 0 and finite static mass density ρ, Willis elasticity reduces [30] to

∇ • (C : ∇u + S • u) -D : ∇u = 0, (4)
where C = C(r) and the two additional rank-3 tensors S = S(r) and D = D(r) generally depend on the spatial position r.

In the case of a homogeneous material or homogenized structure with ∇S i jk = ∇D i jk = 0, which is the focus of interest in this paper, Eq. ( 4) reduces to

∇ • (C : ∇u + W • u) = 0, (5)
with the rank-3 tensor W defined by

W = S -D T , ( 6
)
where the components of the "transposed" tensor are given by

D T i jk = D jik . ( 7
)
Broken centrosymmetry is a necessary requirement for chirality [START_REF] Authier | International Tables for Crystallography Volume D: Physical Properties of Crystals[END_REF]7,[START_REF] Eringen | Elastodynamics[END_REF]]. If we nevertheless consider an isotropic medium or a cubic crystal with centrosymmetry, it follows that W ≡ 0, just like for any homogeneous rank-3 tensor [START_REF] Authier | International Tables for Crystallography Volume D: Physical Properties of Crystals[END_REF]. W ≡ 0 also holds true for an isotropic medium without centrosymmetry.

For a cubic crystal without centrosymmetry, we find that the tensor W , such as any rank-3 tensor

Here, α has SI units of Pa/m. This allows us to rewrite Willis elasticity [START_REF] Milton | Extending the Theory of Composites to Other Areas of Science[END_REF] to

∇ • (C : ∇u) -α ∇ × u = 0. (9)
It is instructive to investigate the behavior of ( 9) under a space-inversion operation, r → -r. As argued below (3), C : ∇u = C : does not change sign, but the ∇ in front does. In the second term in [START_REF] Eringen | Elastodynamics[END_REF], both ∇ and u do change sign, and hence the exterior product does not change sign. Therefore, the relative sign of the first and second term in (9) changes when performing a space inversion. Thus, ( 9) is different for a left-and a right-handed medium, respectively. This behavior is a necessary condition for a continuum formulation to be able to describe the effects of chirality in mechanics. Clearly, if the single parameter beyond linear Cauchy elasticity is zero, α = 0, Eq. ( 9) reduces to linear Cauchy elasticity (3). As usual [START_REF] Banerjee | An Introduction to Metamaterials and Waves in Composites[END_REF][START_REF] Authier | International Tables for Crystallography Volume D: Physical Properties of Crystals[END_REF], for cubic symmetry (with or without a center of inversion), the linear Cauchy elasticity tensor C contains three independent nonzero scalar parameters [START_REF] Authier | International Tables for Crystallography Volume D: Physical Properties of Crystals[END_REF].

We will use [START_REF] Eringen | Elastodynamics[END_REF] for the numerical calculations presented in Sec. III. To be unambiguous and clear for experimentalists, we therefore write Eq. ( 9) out into its three components and explicitly write out all involved sums, leading to

3 i,k,l=1 ∂ ∂x i C i1kl ∂u l ∂x k -α ∂u 3 ∂x 2 - ∂u 2 ∂x 3 = 0, ( 10
) 3 i,k,l=1 ∂ ∂x i C i2kl ∂u l ∂x k -α ∂u 1 ∂x 3 - ∂u 3 ∂x 1 = 0, ( 11
)
and 3 i,k,l=1 ∂ ∂x i C i3kl ∂u l ∂x k -α ∂u 2 ∂x 1 - ∂u 1 ∂x 2 = 0. ( 12
)

III. NUMERICAL CALCULATIONS

In what follows, we illustrate the generalized static homogeneous equation ( 9) for a cubic three-dimensional chiral medium by example numerical calculations. To allow for a direct comparison with micropolar Eringen elasticity, we reproduce previous continuum results [7] and results of metamaterial microstructure calculations [7] at selected points for convenience of the reader. To ease this comparison, we also choose similar parameters as much as possible.

A. Numerical approach

In our numerical calculations, we consider cuboid-shaped samples with volume L × L × 2L. We apply uniaxial loading by a rigid stamp along the negative z direction with sliding boundary conditions at the top, i.e., at the top surface we have a predescribed and fixed nonzero z component of the displacement vector, u z = 0. The components u x and u y can change freely. The axial strain results from = 33 = -u z /L. The uniaxial pressure P exerted at the top is given by P = n • (C : ∇u -α ∇ × u), where n is the normal vector pointing into the negative z direction. On the four sides, we use open boundary conditions, i.e., n •(C : ∇u -α ∇ × u)= 0, with the respective normal vectors n of the four side facets. On the bottom of the cuboid, we use fixed boundary conditions with u = (0, 0, 0) T , describing that the sample cuboid is clamped to a substrate. We have used the same conceptual boundary conditions in our previous work on static Eringen elasticity [7].

We solve Eq. ( 9) by using a finite-element approach via the partial-differential equation (PDE) module of the commercial software package COMSOL MULTIPHYSICS. Herein, the homogeneous sample cuboid is typically discretized into 10 4 tetrahedra, corresponding to about 5 × 10 4 degrees of freedom. Finer discretization has led to negligible changes with respect to the results outlined in the following.

In Sec. III B, we will discuss the behavior of the twist angle and the axial strain of the cuboid sample under uniaxial loading. The axial strain is defined as the z component of the displacement vector at the top surface [which is the same for all positions (x, y, 2L) T ], divided by the sample length 2L, i.e., by u z (x, y, 2L)/(2L).

The twist angle is defined via the displacement of the equivalent four corners at the top of the sample cuboid in the xy plane, which are at positions (±L/2, ±L/2, 2L) T before loading, with respect to the sample center at (0, 0, 2L) T . For pure twists without further deformations, this definition grasps the entire sample behavior. If deformations that are more complex occur in addition, the twist angle resulting from our definition should be seen as merely a parameter representing part of the overall behavior. From such raw data, the twist angle at the top per axial strain can be deduced and plotted vs α (see Fig. 3). All deformations are magnified twofold for clarity. The results of Willis generalized linear Cauchy elasticity in Fig. 1(c) are compared with those of micropolar Eringen elasticity [7] in Fig. 1(b) and finite-element metamaterial microstructure calculations [7] in Fig. 1(a). For the details underlying Figs. 1(a) and 1(b), we refer the reader to the extensive discussion in [7] and the corresponding supporting online material. Obviously, Willis generalized linear Cauchy elasticity, micropolar Eringen elasticity, and the finite-element microstructure calculations exhibit the same qualitative behavior. When replacing α → -α, the direction of the twist changes from clockwise to counterclockwise in Figs. 1(b) and 1(c) (not depicted), corresponding to the behavior of the mirror image of the three-dimensional (3D) microstructure shown in Fig. 1(a).

B. Results and discussion

In Fig. 2(a), the parameters are the same as in Fig. 1(c), except that we consider the three choices (a) α = 3 GPa/m, (b) α → 3.3 α, and (c) α → 33 α. In Fig. 2(b), the twist effect is simply larger than that in Fig. 2(a). In Fig. 2(c), however, unusual additional substructures appear in the displacement field.

Figure 3 emphasizes essentially the same aspect as Fig. 2; however, we do not depict displacement fields of a sample but rather plot the calculated twist/strain (defined in Sec. III A) versus the parameter α for fixed sample side length L. For small values of α, the twist/strain increases monotonously. However, for larger values of α, we find an unusual nonmonotonous resonancelike behavior (compare [19]), which is connected to the behavior shown in Fig. 2(c).

This behavior versus the parameter α for fixed sample side length L is connected to the behavior versus L for fixed α. Following Refs. [7,[START_REF] Eringen | Elastodynamics[END_REF], the effects beyond static linear Cauchy elasticity should decrease with increasing L. More specifically, the twist/strain should decrease proportionally to the surface-to-volume ratio, i.e., decrease ∝1/L. What one gets from (9) versus L for fixed α is the polar opposite of this behavior. This can be seen as follows: If we replace the spatial components x i → ζ x i in [START_REF] Eringen | Elastodynamics[END_REF], with some dimensionless scaling factor ζ , the ratio of the second and first terms in (9) increases by a factor ζ . This means that the effects beyond linear Cauchy elasticity would increase with increasing sample side length L if α was constant. We conclude that α must not be considered as a constant material parameter, but rather as an effective continuum-model parameter. We can equivalently say that the tensor W can be constant and nonzero for a (very) large homogeneous material, but it is zero in the limit of an infinitely large homogeneous material.

To arrive at a meaningful material parameter β, we make the ansatz

α = β L 2 . ( 13
)
The parameter β has SI units of Pa m. Thus, the ratio

l c = β C ( 14
)
has units of a length. Here, C is a nonzero element of the elasticity tensor C or a combination of elements. As the twist effect mainly changes the shape of the specimen but not its volume, we choose the shear modulus C = C 44 = C 2323 . The length l c is obviously zero in the Cauchy limit of α = β = 0. Therefore, it is tempting to interpret l c as a characteristic length scale in the same spirit as characteristic length scales in micropolar Eringen elasticity [START_REF] Eringen | Elastodynamics[END_REF]. There, one gets several different characteristic length scales, all of which are zero in the Cauchy limit.

To test this ansatz for the characteristic length scale, we depict as the upper horizontal scale in Fig. 3 the normalized characteristic length l c /L, which follows from

l c L = β/C 44 L = α L C 44 . ( 15
)
We find that nonmonotonous behavior in Fig. 3 Finally, we study the dependence of the behavior on the sample side length L for fixed parameter β and fixed elements of the linear Cauchy elasticity tensor C (hence, fixed l c ) in Fig. 4. In Fig. 4(a), we plot the twist/strain versus sample side length L on a double-logarithmic scale and in Fig. 4(b) the effective Young's modulus E versus L on a semilogarithmic scale. The twist/strain in Fig. 4(a) decreases inversely proportional to L in the limit L l c (compare dashed straight line). The Young's modulus in Fig. 4(b) initially increases until it reaches a constant level for large values of L. Thereby, linear Cauchy elasticity, for which the twist is zero and the Young's modulus is independent on sample side length, is recovered in the large-sample limit of L → ∞ in Figs. 4(a This overall behavior is qualitatively closely similar to the one which we have recently found in numerical calculations on chiral micropolar Eringen elasticity [START_REF] Eringen | Elastodynamics[END_REF] as well as in our experiments on chiral three-dimensional metamaterials [7], all of which have been performed under static conditions. However, the quantitative agreement with experiments is worse for Willis elasticity than for Eringen elasticity. For example, the effective Young's modulus E versus sample side length L increased by about a factor of ten in [7] before it reached a constant value, whereas it only increases by about 10% before it reaches a constant value in Fig. 4(b). Moreover, the displacement field for the three-dimensional microstructure in Fig. 1

IV. CONCLUSION

In conclusion, we have considered the static version of a generalized form of linear Cauchy elasticity, the Willis equations, for the case of three-dimensional homogeneous chiral noncentrosymmetric cubic media, which have been the subject of recently published experimental and numerical work on mechanical metamaterials. We have found that this form of generalized static linear Cauchy elasticity grasps the quintessential qualitative features of recent experiments and of chiral micropolar Eringen elasticity; however, with just a single additional parameter. Under the same conditions, Eringen elasticity comprises nine additional parameters with respect to linear Cauchy elasticity, three of which directly influence chiral effects.

Such nonuniqueness of effective medium models is common for advanced continuum descriptions of materials in mechanics [34], but also in electromagnetism and optics. It will be interesting to see in the future inasmuch as Willis elasticity is able to describe more advanced static experiments or aspects of dynamic wave propagation in experiments on threedimensional chiral mechanical metamaterials, and whether or not distinct qualitative differences with respect to micropolar Eringen elasticity arise.

  FIG. 1. Cuboid beams with volume L × L × 2L, with side length L = 500 μm, are subject to uniaxial loading along the negative z direction. In linear Cauchy elasticity, the beam compresses and can expand or contract laterally, depending on its Poisson ratio (not depicted). However, a twist is forbidden in linear Cauchy elasticity, even if the underlying crystal symmetry would allow for a twist. (a) Finite-element calculation for a chiral metamaterial microstructure exhibiting a twist behavior [7]. (b) Same as (a), but calculated using chiral micropolar Eringen elasticity [7]. (a) and (b) are taken with permission from [7]. (c) Numerical calculations based on generalized linear Cauchy elasticity following Willis. For all three panels, the effects are calculated within the linear elastic regime and are magnified tenfold for clarity. The modulus of the displacement vector field is superimposed on a false-color scale. Parameters in (c) are C 11 = 32.8 MPa, C 12 = -6.1 MPa, C 44 = 19.4 MPa (in Voigt notation), and α = 3 GPa/m.

  [3], reduces to the rank-3 Levi-Civita tensor ε (with components ε 123 = ε 231 = ε 312 = -ε 132 = -ε 213 = -ε 321 = 1, all other components are zero) times a scalar factor α, i.e., W = α ε.(

FIG. 2 .

 2 FIG. 2. (a) Same as Fig. 1(c) with parameter α = 3 GPa/m and all other parameters fixed as in Fig. 1(c). (b) α → 3.3 α. (c) α → 33 α.From such raw data, the twist angle at the top per axial strain can be deduced and plotted vs α (see Fig.3). All deformations are magnified twofold for clarity.

Figure 1

 1 Figure 1 shows the modulus of the displacement vector field (on a false-color scale) for uniaxial loading along the negative z axis of a cuboid-shaped sample with volume L × L × 2L. All results shown are within the linear elastic regime, i.e., for axial strains <1%. We choose L = 500 μm (see N = 1 in [7]), C 11 = C 22 = C 33 (in standard Voigt notation [9]), C 12 = C 13 = C 21 = C 23 = C 31 = C 32 , C 44 = C 55 = C 66 (all other elements of the elasticity tensor are zero), and α = 3 GPa/m. The latter value has been chosen to match the experiments (and hence also the results of the microstructure calculations and those of Eringen continuum elasticity; cf. [7]).The results of Willis generalized linear Cauchy elasticity in Fig.1(c) are compared with those of micropolar Eringen elasticity[7] in Fig.1(b) and finite-element metamaterial microstructure calculations[7] in Fig.1(a). For the details underlying Figs.1(a) and 1(b), we refer the reader to the extensive discussion in[7] and the corresponding supporting online material. Obviously, Willis generalized linear Cauchy elasticity, micropolar Eringen elasticity, and the finite-element microstructure calculations exhibit the same qualitative behavior. When replacing α → -α, the direction of the twist changes from clockwise to counterclockwise in Figs.1(b) and 1(c) (not depicted), corresponding to the behavior of the mirror image of the three-dimensional (3D) microstructure shown in Fig.1(a).In Fig.2(a), the parameters are the same as in Fig.1(c), except that we consider the three choices (a) α = 3 GPa/m, (b) α → 3.3 α, and (c) α → 33 α. In Fig.2(b), the twist effect is simply larger than that in Fig.2(a). In Fig.2(c), however, unusual additional substructures appear in the displacement field.Figure3emphasizes essentially the same aspect as Fig.2; however, we do not depict displacement fields of a sample but rather plot the calculated twist/strain (defined in Sec. III A) versus the parameter α for fixed sample side length L. For small values of α, the twist/strain increases monotonously.

FIG. 3 .

 3 FIG.3. Twist/strain vs parameter α (lower horizontal scale) as deduced from raw data such as the ones shown in Fig.2. The dots are calculated; the curve is a guide to the eye. Parameters are as in Fig.1(c). The upper horizontal scale shows the characteristic length l c normalized to the sample side length L, lc L = β/C 44 L = α L C 44 , according to[START_REF] Seppecher | [END_REF]. Here, C 44 is an element of the elasticity tensor in Voigt notation, namely, the shear modulus C 2323 .

  occurs when l c becomes comparable to or even exceeds the sample side length L. Likewise, the characteristic length l c = 1275 μm in Fig.2(c) is also larger than the sample side length of L = 500 μm, whereas l c is smaller by factor of 33 and 10, respectively, in Figs. 3(a) and 3(b).

FIG. 4 .

 4 FIG. 4. (a) Twist/strain vs sample side length L normalized by the characteristic length l c on a double-logarithmic scale. The dependence on the parameter β is implicitly contained in this normalization. The dashed straight line has a slope of -1, corresponding to the expected asymptotic scaling of the twist angle ∝1/L for fixed parameter β. (b) Same as in (a), but for the effective Young's modulus E on a semilogarithmic scale. All other parameters are as in Fig. 1(c). The dots are calculated; the curves are guides to the eye.

  ) and 4(b)-as it should.

  (a), which agrees quantitatively with Eringen elasticity in Fig. 1(b), shows a somewhat more pronounced minimum in the middle of the sample top facet for Willis elasticity in Fig. 1(c).

ACKNOWLEDGMENTS

We thank Graeme W. Milton, Jensen Li, and Christian Kern (KIT) for stimulating discussions. We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy via the Excellence Cluster 3D Matter Made to Order (EXC-2082 390761711), by the Carl Zeiss Foundation through the "Carl-Zeiss-Focus@HEiKA", by the Helmholtz program "Science and Technology of Nanosystems" (STN), and by the associated KIT project "Virtual Materials Design" (VIRT-MAT). M.K. acknowledges support by the EIPHI Graduate School (Contract No. ANR-17-EURE-0002) and the French Investissements d'Avenir program, project ISITE-BFC (Contract No. ANR-15-IDEX-03).