Romain Jolivet
email: romain.jolivet@ens.fr

M Simons

Z Duputel

J.-A Olive

H S Bhat

& Q Bletery

Interseismic loading of subduction megathrust drives long term uplift in northern Chile

We propose an approach to quantify the ratio between elastic and inelastic deformation in subduction forearcs

• Two distinct correlations can be observed between interseismic and quaternary uplift rates

• We propose that 4 to 8% of interseismic uplift rates translates into persistent deformation in Northern Chile

Introduction

Along a subduction interface, the quasi-continuous motion of converging plates leads to the build up of elastic stress where the interface is locked [START_REF] Savage | A Dislocation Model of Strain Accumulation and Release at a Subduction Zone[END_REF]. Large earthquakes, occurring as slip on the megathrust interface, redistribute elastic energy accumulated within the surrounding medium. In addition to these sudden changes in stress, parts of the subduction interface may slip aseismically during the interseismic period, either episodically during slow slip events [START_REF] Hirose | A slow thrust slip event following the two 1996 Hyuganada Earthquakes beneath the Bungo Channel, southwest Japan[END_REF][START_REF] Dragert | A Silent Slip Event on the Deeper Cascadia Subduction Interface[END_REF] or over decades to centuries [START_REF] Mazzotti | Full interseismic locking of the Nankai and Japan-west Kurile subduction zones: An analysis of uniform elastic strain accumulation in Japan constrained by permanent GPS[END_REF][START_REF] Metois | Interseismic Coupling, Megathrust Earthquakes and Seismic Swarms Along the Chilean Subduction Zone[END_REF]. Averaged over the interseismic period, a coupling coefficient, γ, is usually used to describe the inferred local slip deficit normalized by the long term convergence rate along the subduction megathrust (Fig. 1). γ = 1 corresponds to a fully locked interface, with a potential for slip during future earthquakes. γ = 0 indicates an inferred slip rate equal to the local plate convergence rate [START_REF] Savage | A Dislocation Model of Strain Accumulation and Release at a Subduction Zone[END_REF]. Subduction interfaces are paved with locked sections (i.e. portions of the interface with γ 1) separated by creeping sections [e.g. [START_REF] Konca | Partial rupture of a locked patch of the Sumatra megathrust during the 2007 earthquake sequence[END_REF][START_REF] Nocquet | Motion of continental slivers and creeping subduction in the northern Andes[END_REF] (i.e. γ << 1).

Large megathrust earthquakes uplift the surface located above a patch of slip and induce subsidence around it [START_REF] Savage | A Dislocation Model of Strain Accumulation and Release at a Subduction Zone[END_REF][START_REF] Vigny | The 2010 Mw[END_REF][START_REF] Simons | The 2011 Magnitude 9.0 Tohoku-Oki Earthquake: Mosaicking the Megathrust from Seconds to Centuries[END_REF]. To the extent that the coupling coefficient, γ, is stable through time, segments that ruptured during a large event reload during the interseismic period causing surface deformation of opposite sign to that occurring during the earthquake. In the most simplistic model, the amount of slip during earthquakes should be balanced by the slip deficit that builds up in the interim period, at least once integrated over many earthquake cycles. However, along several subduction zones, the coastal domain, a region defined to extend from the coast to the continental shelf (Fig. 1), experiences long-lived vertical displacement, either subsidence or uplift, over geological times (i.e. 10 5 -10 6 years), for example as inferred from the geometry of raised terraces in Chile [START_REF] Saillard | Non-steady long-term uplift rates and Pleistocene marine terrace development along the Andean margin of Chile (31°S) inferred from 10Be dating[END_REF][START_REF] Regard | Renewed uplift of the Central Andes Forearc revealed by coastal evolution during the Quaternary[END_REF], Japan [START_REF] Matsu'ura | Late Quaternary uplift rate inferred from marine terraces, Muroto Peninsula, southwest Japan: Forearc deformation in an oblique subduction zone[END_REF] or Greece [START_REF] Mouslopoulou | Uplift rate transients at subduction margins due to earthquake clustering[END_REF] or the erosion patterns along rivers in Mexico [START_REF] Ramírez-Herrera | Relating the long-term and short-term vertical deformation across a transect of the forearc in the central Mexican subduction zone[END_REF]. Such long-term, irrecoverable deformation can be modeled with a purely plastic rheology, for instance using critical taper theory [START_REF] Davis | Mechanics of Fold-and-Thrust Belts and Accretionary wedges[END_REF]. Such models, however, cannot address the question of when inelastic deformation cumulates in the forearc with respect to the different phases of the earthquake cycle.

For comparison, typical vertical displacement rates over geological times for the coastal domain are generally one order of magnitude lower than typical interseismic uplift or subsidence rates [a fraction of millimeter per year vs several millimeters per year, e.g. [START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF][START_REF] Jolivet | A Multipixel Time Series Analysis Method Accounting for Ground Motion, Atmospheric Noise, and Orbital Errors[END_REF][START_REF] Regard | Renewed uplift of the Central Andes Forearc revealed by coastal evolution during the Quaternary[END_REF][START_REF] Melnick | Rise of the central Andean coast by earthquakes straddling the Moho[END_REF]Hashima and Sato, 2017, Fig. 1]. Comparatively, large megathrust earthquakes will generate meter-scale vertical displacements every time they occur [e.g. [START_REF] Vigny | The 2010 Mw[END_REF][START_REF] Simons | The 2011 Magnitude 9.0 Tohoku-Oki Earthquake: Mosaicking the Megathrust from Seconds to Centuries[END_REF]. We cannot yet assess how much permanent uplift or subsidence occurs during earthquakes [START_REF] Baker | Permanent deformation caused by subduction earthquakes in northern Chile[END_REF][START_REF] Melnick | Rise of the central Andean coast by earthquakes straddling the Moho[END_REF] or during the interseismic period [START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF][START_REF] Saillard | From the seismic cycle to long-term deformation: linking seismic coupling and Quaternary coastal geomorphology along the Andean megathrust[END_REF], as our understanding of the earthquake cycle is limited to simplified elastic and visco-elastic models which do not allow for permanent deformation to build up within the upper crust.

Béjar

Focusing on the interseismic period, correlations between distributions of coupling coefficients along megathrusts and morphotectonic features [START_REF] Victor | Long-term persistence of subduction earthquake segment boundaries: Evidence from Mejillones Peninsula, northern Chile[END_REF][START_REF] Saillard | From the seismic cycle to long-term deformation: linking seismic coupling and Quaternary coastal geomorphology along the Andean megathrust[END_REF] or large amplitude gravity anomalies [START_REF] Song | Large trench-parallel gravity variations predict seismogenic behavior in subduction zones[END_REF]Wells et al., 2003] suggest that interseismic fault locking is to first order a long-term feature, stable over many seismic cycles. Interseismic fault locking is thought to imprint its signature in the morphology of the subduction forearc. For instance, a qualitative relationship has been described between the maximum depth of high coupling coefficient along subduction megathrust and the position of the coastal domain [START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF][START_REF] Rousset | Lateral Variations of Interplate Coupling along the Mexican Subduction Interface: Relationships with Long-Term Morphology and Fault Zone Mechanical Properties[END_REF][START_REF] Mouslopoulou | Uplift rate transients at subduction margins due to earthquake clustering[END_REF][START_REF] Saillard | From the seismic cycle to long-term deformation: linking seismic coupling and Quaternary coastal geomorphology along the Andean megathrust[END_REF] (Fig. 1). If part of the uplift measured during the interseismic period is not recovered elastically during earthquakes occurring along the megathrust, then anelastic deformation accumulates and the coastal domain will uplift over geological times [START_REF] Song | Large trench-parallel gravity variations predict seismogenic behavior in subduction zones[END_REF][START_REF] Mouslopoulou | Uplift rate transients at subduction margins due to earthquake clustering[END_REF][START_REF] Saillard | From the seismic cycle to long-term deformation: linking seismic coupling and Quaternary coastal geomorphology along the Andean megathrust[END_REF]. Because most of the slip is often located offshore, earthquakes along subduction zones tend to lower the coastal domain [START_REF] Simons | The 2011 Magnitude 9.0 Tohoku-Oki Earthquake: Mosaicking the Megathrust from Seconds to Centuries[END_REF][START_REF] Duputel | The Iquique earthquake sequence of April 2014: Bayesian modeling accounting for prediction uncertainty[END_REF]. However, if coseismic slip reaches depths below the coastline, the coastal domain will uplift coseismically [START_REF] Grandin | Three-dimensional displacement field of the 2015 Mw8.3 Illapel earthquake (Chile) from across-and along-track Sentinel-1 TOPS interferometry[END_REF][START_REF] Vigny | The 2010 Mw[END_REF].

The residual permanent vertical displacement will therefore depend on the relative balance between coseismic slip and interseismic slip deficit and on the position of the hinge line with respect to the coastal domain [START_REF] Saillard | From the seismic cycle to long-term deformation: linking seismic coupling and Quaternary coastal geomorphology along the Andean megathrust[END_REF]. Contradictory conclusions have been reached considering the contribution of earthquakes and interseismic loading to the permanent deformation of the forearc. For instance, [START_REF] Melnick | Rise of the central Andean coast by earthquakes straddling the Moho[END_REF] proposes that intermediate-depth earthquakes are responsible for the uplift of the northern Chilean coast while he discards the influence of interseismic fault locking. Other studies point to spatial relationships between patterns of coupling coefficients and long term features of the forearc in support of the influence of interseismic fault locking on topography building [START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF][START_REF] Rousset | Lateral Variations of Interplate Coupling along the Mexican Subduction Interface: Relationships with Long-Term Morphology and Fault Zone Mechanical Properties[END_REF][START_REF] Mouslopoulou | Uplift rate transients at subduction margins due to earthquake clustering[END_REF][START_REF] Saillard | From the seismic cycle to long-term deformation: linking seismic coupling and Quaternary coastal geomorphology along the Andean megathrust[END_REF].

Data and Methods: The case of northern Chile

Here, we use direct estimates of interseismic displacement rates and compare these with long term uplift rates in subduction regions. Geodetic measurements of surface displacement rates, including GNSS (historically referred to as GPS) and InSAR (Synthetic Aperture Radar Interferometry) measurements, are available over many subduction forearcs. InSAR provides measurements of surface deformation along the direction of the Line-Of-Sight (LOS) of a satellite. For SAR acquisitions from the Envisat satellite, for instance, the LOS incidence angle is approximately 23 • resulting in a high sensitivity to vertical displacements. However, InSAR is also sensitive to horizontal displacements confounding attempts to isolate a purely vertical component of displacement. GNSS provides time series of three-component displacements at sparsely distributed locations. However, uncertainties on GNSS-derived vertical displacement rates typically reach several millimeters per year, making it a challenge to measure the sub-millimetric displacement rates expected from permanent deformation [START_REF] Melnick | Rise of the central Andean coast by earthquakes straddling the Moho[END_REF].

One solution is to estimate a model of the distribution of coupling coefficients along the megathrust that captures all measurements of vertical, horizontal and LOS displacements along with their respective uncertainties in order to produce an a posteriori estimate of vertical interseismic motion, and compare those predictions with long-term displacement rates. Along subduction zones, where locking of the megathrust is mostly offshore, it is possible to infer locked asperities with geodetic measurements made onshore [START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF][START_REF] Lin | High interseismic coupling in the Eastern Makran (Pakistan) subduction zone[END_REF]. However, because we are generally limited to observations made at the surface and onshore, inferred distributions of coupling coefficient often differ significantly due to different modeling strategies and different regularizations of the inverse problem. In the present study, we do not particularly care about this nonuniqueness since we only use these models to provide the best possible estimate of vertical displacement rates over the interseismic period consistent with the available geodetic data. Effectively, models are used to interpolate the data.

Considering the duration of the modern instrumental period with respect to both the characteristic time of the earthquake cycle and the longer time required to accumulate significant anelastic deformation, it is crucial to focus on a region where we will be able to measure surface displacements that are unambiguously related to interseismic loading of the megathrust. In particular, the relaxation period following a large earthquake might extend over decades and could strongly influence what is considered as interseismic deformation [START_REF] Trubienko | Interpretation of interseismic deformations and the seismic cycle associated with large subduction earthquakes[END_REF][START_REF] Hashima | A megathrust earthquake cycle model for Northeast Japan: bridging the mismatch between geological uplift and geodetic subsidence[END_REF]. Finally, this region must be densely sampled by GNSS measurements and its geographical characteristics must be optimal for InSAR.

Northern Chile has been hit by several earthquakes, including the large, M w 7.7, Tocopilla earthquake [START_REF] Bejar-Pizarro | Asperities and barriers on the seismogenic zone in North Chile: state-of-the-art after the 2007 Mw 7.7 Tocopilla earthquake inferred by GPS and InSAR data[END_REF][START_REF] Schurr | The 2007 M7.7 Tocopilla northern Chile earthquake sequence: Implications for along-strike and downdip rupture segmentation and megathrust frictional behavior[END_REF] in 2007 and the great, M w 8.1, Iquique earthquake [START_REF] Duputel | The Iquique earthquake sequence of April 2014: Bayesian modeling accounting for prediction uncertainty[END_REF] in 2014. There, inferences of alongstrike variations of coupling coefficients are broadly consistent between models [START_REF] Metois | Interseismic Coupling, Megathrust Earthquakes and Seismic Swarms Along the Chilean Subduction Zone[END_REF][START_REF] Schurr | Gradual unlocking of plate boundary controlled initiation of the 2014 Iquique earthquake[END_REF]. The last great megathrust event prior to the 2014 Iquique earthquake occurred in 1877 with an estimated magnitude of 8.6 [START_REF] Comte | Reappraisal of great historical earthquakes in the northern Chile and southern Peru seismic gaps[END_REF][START_REF] Metois | Revisiting the North Chile seismic gap segmentation using GPS-derived interseismic coupling[END_REF]. Therefore, we assume that surface displacement rates in this region prior to the 2014 Iquique earthquake are not significantly changing due to postseismic viscous relaxation and truly reflect the effect of locking along the megathrust.

The relative aridity of northern Chile makes it an ideal target for InSAR measurements [e.g. Pritchard and Simons, 2002;[START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF]. We use SAR data acquired by the Envisat satellite over the 2003-2010 period to derive a map of LOS ground velocity (Fig. 2). We use a multi-pixel method to infer displacement time series, displacement rates averaged over the observation period and earthquake-related offsets accounting for spatial covariances as well as other nuisance parameters [START_REF] Jolivet | A Multipixel Time Series Analysis Method Accounting for Ground Motion, Atmospheric Noise, and Orbital Errors[END_REF].

Our approach resolves long wavelength signals in SAR acquisitions, hence does not require input from GNSS data. The resulting map of LOS displacement rates highlights a variable uplift rate along the coastal domain, with LOS rates spanning a 0-4 mm/yr range.

Measurements of horizontal interseismic displacement rates are available thanks to the dense continuous GNSS measurements that have been deployed since 2000 [START_REF] Simons | Central Andean Tectonic Observatory Geodetic Array[END_REF] and are complementary to our InSAR displacement rate maps [START_REF] Metois | Interseismic Coupling, Megathrust Earthquakes and Seismic Swarms Along the Chilean Subduction Zone[END_REF].

We use a backslip approach to estimate the coupling coefficient from geodetic displacement rates [START_REF] Savage | A Dislocation Model of Strain Accumulation and Release at a Subduction Zone[END_REF] and apply a Bayesian formalism to explore the range of possible models given the GNSS and InSAR surface rates [START_REF] Jolivet | Aseismic slip and seismogenic coupling along the central San Andreas Fault[END_REF]. The mean interseismic coupling model is broadly consistent with the most recent published studies [START_REF] Li | Revisiting viscoelastic effects on interseismic deformation and locking degree: A case study of the Peru-North Chile subduction zone[END_REF][START_REF] Metois | Interseismic Coupling, Megathrust Earthquakes and Seismic Swarms Along the Chilean Subduction Zone[END_REF]. We infer a highly coupled segment extending northwards from, at least, Antofagasta in the south up to 20.5 • S and a relatively less coupled segment offshore Iquique (Fig. 3). The mean model has lower coupling coefficients at the trench but this feature is not resolved given that we are limited to observations made onshore (see supplementary materials). There is a clear and robust separation between the coupled segments, with a narrow barrier with a coupling coefficient of 0 (maximum a posteriori, mean equals 0.1 ± 0.1). Contrary to recently published models [START_REF] Li | Revisiting viscoelastic effects on interseismic deformation and locking degree: A case study of the Peru-North Chile subduction zone[END_REF][START_REF] Metois | Interseismic Coupling, Megathrust Earthquakes and Seismic Swarms Along the Chilean Subduction Zone[END_REF], we do not infer fault locking underneath the south American continent. This inference of no kinematic locking extending below the coastline is driven by the InSAR observations indicating only onshore uplift [START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF].

Our goal is to compare the short-term vertical displacements predicted by our coupling models to long-term estimates of coastal uplift. Figure 4 shows the vertical displacement rates over the forearc predicted by our mean coupling model. To first order, topography shows striking similarities with this pattern of vertical rates, as maximum uplift is located in the coastal cordillera. However, topography is a signal integrated over millions of years, amalgamating the effects of tectonics, mantle dynamics and erosion. Therefore, topography alone cannot be simply used to study the influence of short term megathrust dynamics on the building of topography.

The presence of marine terraces all along the chilean coast documents the long term uplift of the coastal domain over the quaternary [START_REF] Regard | Renewed uplift of the Central Andes Forearc revealed by coastal evolution during the Quaternary[END_REF][START_REF] Saillard | From the seismic cycle to long-term deformation: linking seismic coupling and Quaternary coastal geomorphology along the Andean megathrust[END_REF].

Using a landscape evolution model that accounts for variations in sea level to reproduce the topography of the coastal domain and, in particular, the shape of the so-called coastal rasa, based on the competition of wave erosion and uniform uplift, [START_REF] Melnick | Rise of the central Andean coast by earthquakes straddling the Moho[END_REF] determined uplift rates over the quaternary with dense spatial sampling. These rates quantify the uplift of the Chilean coast at an average rate of 0.13 ± 0.04 mm/yr since, at least, 1 Myr.

Lateral variations within these quaternary uplift rates are visible along the coast.

Melnick [2016] finds no obvious correlation between the long-term rates and interseismic rates. However, the interseismic model used to predict interseismic rates assumes fully homogeneous kinematic locking of the megathrust down to 35 km-depth without along-strike variations, which is inconsistent with our observations and model of coupling coefficient (Fig. 3) as well as with some previously published models [e.g. [START_REF] Li | Revisiting viscoelastic effects on interseismic deformation and locking degree: A case study of the Peru-North Chile subduction zone[END_REF][START_REF] Metois | Interseismic Coupling, Megathrust Earthquakes and Seismic Swarms Along the Chilean Subduction Zone[END_REF].

Results: comparing short and long term uplift rates

In contrast to previous analysis, we observe a slight correlation (Pearson productmoment correlation coefficient of 0.5) between the distribution of coupling coefficients offshore (i.e. 50 km offshore in the direction of convergence) and quaternary uplift of the coastal domain (Fig. 4). Quaternary uplift appears faster where coupling coefficient is high along the megathrust. However, because of the inherent limitations in inferred models of coupling (i.e. non-uniqueness), we rather rely on the relationship between predicted interseismic and long-term uplift. Effectively, as alluded to earlier, we consider our coupling model as no more than a physics-based interpolation, including uncertainties, of interseismic surface displacements along the coast. Although we do not include vertical GNSSderived velocities in our inversion, a first order comparison of the predicted interseismic uplift at available GNSS stations suggests a relatively correct agreement with our predictions [see supp. mat. and [START_REF] Blewitt | MIDAS robust trend estimator for accurate GPS station velocities without step detection[END_REF]. Such validation against an independent data set should be taken with caution as formal uncertainties in GNSS-derived velocities might be underestimated, especially for vertical rates of motion. In addition, we need to consider that numerous active faults have been recognized in the forearc, in particular over the regions of Tocopilla, Salar Grande and North of Pisagua [START_REF] Allmendinger | Invited review paper: Neogene to Quaternary tectonics of the coastal Cordillera, northern Chile[END_REF]. All types of faulting, reverse, normal and strike slip, active over the neogene and quaternary have been identified in specific regions along the coastline, which could indicate along strike variations in the long term behavior of the forearc.

Plotting the interseismic, predicted, uplift rates with respect to the modeled quaternary uplift rates, it appears that points can be separated in two groups (please refer to supplementary materials for a plot without our interpretation). Within each group, short and long term rates correlate. After separating these points in two distinct groups, we find that there is a spatial coherence within these groups. In a first group (colored dots on figure 4), we consider points from regions that have experienced extensive neogene and quaternary active faulting according to [START_REF] Allmendinger | Invited review paper: Neogene to Quaternary tectonics of the coastal Cordillera, northern Chile[END_REF], while in the second group, we consider points from regions devoid of active faulting (black dots in Fig. 4).

Points are not randomly taken in a group or the other and rather cluster spatially along the coastline. Taken independently, the correlation between short-term and quaternary uplift shows correlation coefficients of 0.74 and 0.76 for the areas with and without recent active faulting, respectively. While the separation might sound ad hoc at first, we believe the spatial coherence of these two groups of points is the signature of an along-strike variable behavior of the forearc subjected to interseismic loading.

For a given interseismic uplift rate, estimates of long term coastal uplift are consistently slower in areas that have experienced significant quaternary faulting. We fit both data sets separately using a Bayesian regression technique (see supp. mat.) and infer the distribution of plausible linear relationships between quaternary and short-term uplift. In both cases, we observe that inferred quaternary uplift rates correspond to 4 to 8% of our estimated interseismic rates in northern Chile. In addition, it appears that, while a zerovalued interseismic uplift rate corresponds to a zero-valued quaternary uplift rate for the regions affected by recent active faulting, this is not the case for regions without identified active faulting. In the later case, a zero-valued interseismic uplift rate corresponds to a 0.1±0.05 mm/yr quaternary uplift rate.

Discussion: Internal anelastic deformation and the role of faulting

We interpret the observed correlation(s) between million-year time scale uplift and present-day interseismic uplift as the long-term signature of interseismic deformation.

We explain the distribution of short vs. long term uplift rates as the combination of three overlapping mechanisms. First, a broad regional scale uplift of 0.1±0.05 mm/yr affects the whole coastline in northern Chile. Second, 4 to 8% of the interseismic uplift is not balanced by co-and post-seismic slip related to large megathrust earthquakes. Third, a systematic offset of about 0.1 mm/yr is observed between regions that have and have not experienced active faulting over the quaternary. Our interpretation therefore suggests that interseismic and coseismic deformations do not cancel each other over many cycles and that approximately 4 to 8% of the vertical interseismic uplift is permanent. While the imprint of fault locking on the forearc morphology has been previously hypothesized based on the trench parallel gravity anomaly [e.g. [START_REF] Song | Large trench-parallel gravity variations predict seismogenic behavior in subduction zones[END_REF]Wells et al., 2003], we provide here a quantitative estimate of the amount of permanent deformation for northern Chile.

Then, two hypotheses could explain the systematic difference of uplift rates in regions with and without recent (neogene to quaternary) active faulting. First, slip along shallow structures within the forearc may have dissipated part of the energy that should have transferred into gravitational potential energy (i.e. topography), hence a lesser efficiency at building topography. Second, the presence of such pervasive faulting could imply variations in forearc material properties along the coast. These hypotheses now need to be tested against in-depth modeling.

Essential to our interpretation is that forearc materials must deform beyond their elastic limit in between large megathrust earthquakes at characteristic interseismic strain rates of 10 -17 to 10 -15 s -1 (i.e. 0.1 to 10 nanostrain per year), and upper crustal temperatures and pressures below ∼ 400 • C and ∼ 200 MPa, respectively. Among possible candidate mechanisms, pressure solution processes enable viscous-like creep under low deviatoric stress and temperature, particularly in shallow porous rock units (e.g., sedimentary layers) [START_REF] Niemeijer | Compaction creep of quartz sand at 400-600°C: experimental evidence for dissolution-controlled pressure solution[END_REF]. Under greater deviatoric stresses, low-porosity rocks are more likely to deform inelastically through brittle creep -the macroscopic manifestation of grain-scale cracks that nucleate from pre-existing defects and grow sub-critically through stress-and temperature-activated chemical processes [START_REF] Paterson | Experimental Rock Deformation -The Brittle Field[END_REF][START_REF] Brantut | Time-dependent cracking and brittle creep in crustal rocks: A review[END_REF].

We illustrate the possible behavior of the forearc through the example of quartz, a material typical of a continental crust, which can deform visco-plastically through dislocation glide at low temperature [START_REF] Bhat | The Micromechanics of Westerley Granite at Large Compressive Loads[END_REF]. Under a compressive load, such material behaves elastically at low strain regime and starts to deviate from elasticity at higher strain and the style of deformation (i.e. elastic or inelastic) depends on the ambient strain rate (Fig. 5). For strain rates on the order of those expected for the interseismic period within the forearc (i.e. 10 -17 to 10 -15 s -1 ), quartz behaves elastically for stresses lower than about 100 MPa, with inelastic strain increasing at higher stresses. Such a visco-plastic material would allow for interseismic loading up to absolute stresses on the order of those of a typical stress drop (i.e. less than 100MPa), while accumulating permanent strain. In addition, because quartz tends to be elastic at high strain rates, we can assume the forearc would behave elastically during the coseismic (and maybe immediate post-seismic) period.

Our interpretation focuses only on the behavior of the upper crust of the overriding plate, and thus does not account for the viscous rheology of the upper mantle over the earthquake cycle [START_REF] Wang | Deformation cycles of subduction earthquakes in a viscoelastic Earth[END_REF], which may affect surface strain rates and estimates of coupling along the megathrust [START_REF] Li | Revisiting viscoelastic effects on interseismic deformation and locking degree: A case study of the Peru-North Chile subduction zone[END_REF]. Also, we do not account for other potential transient deformation events. Post-seismic stress relaxation following large earthquakes, for instance, occurs through aseismic slip [e.g. [START_REF] Hsu | Frictional Afterslip Following the 2005 Nias-Simeulue Earthquake, Sumatra: Supplementary material[END_REF] and viscous flow within the upper-mantle [e.g. [START_REF] Trubienko | Interpretation of interseismic deformations and the seismic cycle associated with large subduction earthquakes[END_REF]. The influence of mantle relaxation considering a crust that can deform inelastically has not been extensively modeled yet. That said, we believe that accounting for viscous processes in the mantle and afterslip would not modify our conclusions based purely on the comparison of surface deformation. First, given the geological time scales required to build topography, we can consider large earthquakes and their corresponding afterslip as single slip events. Second, post-seismic viscous flow within the upper-mantle following a large earthquake puts the forearc in compression for a period of time that depends on the characteristic time involved [START_REF] Sun | Prevalence of viscoelastic relaxation after the 2011 Tohokuoki earthquake[END_REF]. If that characteristic time is small with respect to large megathrust cycles, it should be negligible.

Other examples worldwide do not necessarily compare simply with the behavior we observe in northern Chile. For instance, in Japan, the coastline of Tohoku subsides both during large earthquakes [e.g. [START_REF] Simons | The 2011 Magnitude 9.0 Tohoku-Oki Earthquake: Mosaicking the Megathrust from Seconds to Centuries[END_REF] and during the interseismic period [e.g. [START_REF] Hu | Stress-driven relaxation of heterogeneous upper mantle and time-dependent afterslip following the 2011 Tohoku earthquake[END_REF]. Only during the post-seismic period, a selection of GNSS-derived ground velocities indicates uplift of the forearc. Although all phases of the earthquake cycle have been recorded in this region, the summation of observed coseismic, modeled postseismic and observed interseismic motion cannot explain the long-term uplift observed in the area.

Visco-elastic models of interseismic motion allow for a potential reversal of rates of vertical motion during the interseismic period [START_REF] Trubienko | Interpretation of interseismic deformations and the seismic cycle associated with large subduction earthquakes[END_REF][START_REF] Hashima | A megathrust earthquake cycle model for Northeast Japan: bridging the mismatch between geological uplift and geodetic subsidence[END_REF], which could reconcile long and short term observations in this part of Japan. In northern Chile, the only available surface deformation measurements are during the interseismic period and for moderate sized earthquakes and their respective post-seismic sequences. Considering the example of Tohoku, surface displacements related to the largest possible event (and its postseismic displacements) along the megathrust completely override signals from other smaller magnitude events. Therefore, a complete budget of vertical displacements is impossible here and we can only hypothesize based on the observed cor- [START_REF] Jolivet | A Multipixel Time Series Analysis Method Accounting for Ground Motion, Atmospheric Noise, and Orbital Errors[END_REF]. Dark red contour lines are 2 m slip contours for earthquakes of magnitude larger than 8, including the Antofagasta, Mw 8.1, 1995 event [Pritchard et al., 2002] and the Iquique, Mw 8.1, 2014 event [START_REF] Duputel | The Iquique earthquake sequence of April 2014: Bayesian modeling accounting for prediction uncertainty[END_REF]. Orange contour lines are 0.5 m slip contours for earthquakes of magnitude between 7 and 8, including the 2007, Mw 7.7, Tocopilla event [START_REF] Béjar-Pizarro | Andean structural control on interseismic coupling in the North Chile subduction zone[END_REF] and the 2014, Mw 7.7, Iquique's biggest aftershock [START_REF] Duputel | The Iquique earthquake sequence of April 2014: Bayesian modeling accounting for prediction uncertainty[END_REF]. Focal mechanisms for earthquakes larger than magnitude 7.5 since 1995 are from the Global CMT project [START_REF] Ekström | The global CMT project 2004-2010: Centroid-moment tensors for 13,017 earthquakes[END_REF].

Topography is from SRTM [START_REF] Farr | Shuttle radar topography mission produces a wealth of data[END_REF].

Figure 1 .Figure 2 .

 12 Figure1. Simplified seismotectonic setting of the subduction forearc -An oceanic plate slides underneath the subduction forearc. The interface is divided between a locked section and a creeping, unlocked section. The coupling coefficient is the parameter describing the apparent slip deficit normalized by the longterm plate convergence rate. A coupling coefficient of 0 indicates slip at plate rate along the interface while a coupling coefficient of 1 indicates full kinematic locking. The coastal domain is the region proximal to the coastline that is submitted to changes in sea level. During the interseismic period, uplift is maximum at the hinge line, which corresponds to the surface projection of the transition between the locked and unlocked sections of the interface at depth.

Figure 4 .

 4 Figure 4. Comparison between short term interseismic fault behavior and quaternary coastal uplift -Map shows quaternary coastal uplift rate [Melnick, 2016] (colored circles) and interseismic uplift rates predicted from our model (background color). (Top right) Quaternary coastal uplift as a function of coupling coefficient offshore, 50 km from the coast in a direction parallel to relative plate motion. (Bottom right) Quaternary coastal uplift as a function of interseismic uplift with our interpretation of the relationship between interseismic and quaternary uplift. Colored dots indicate locations where quaternary active faulting has been identified [please refer to the map for the location of these colored points; Allmendinger and González, 2010]. Thick blue and black lines are the linear trends predicted from the Bayesian regression for the region where quaternary active faulting has been identified and not identified, respectively. Thin lines show a hundred realizations of the posterior PDF derived by the Bayesian regression, reflecting the range of possible trends for each group of points. See supplementary materials for a similar plot without interpretation, for a figure highlighting the position of the various points on a map and for a description of the regression applied.

Acknowledgements

This project has received funding from NASA (grant NNX16AK58G). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreements 758210 and 805256). This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the program Investissements d'Avenir supervised by the Agence Nationale pour la Recherche. Envisat raw data have been obtained upon request via the EOLISA tool. We thank the European Space Agency for the acquisition and the distribution of these data.

ERA-Interim products are directly available for download at ECMWF (https://www.ecmwf.int/).

The authors are grateful for enlightening discussions with J.-P. Avouac, N. Cubas, L. Dal Zilio and N. Brantut. The authors thank the editor and both an anonymous reviewer and R. Bürgmann for their constructive comments that helped crafting a better manuscript.

rate dependent, we show the corresponding behavior for 4 strain rates corresponding to a range of interseismic strain rates. (Right) Schematic behavior of a visco-plastic material to the cycles of loading and unloading imposed by the succesion of interseismic and coseismic phases on the megathrust. During the interseismic phase, stress increases in the forearc, at a strain rate imposed by the slip rate on the megathrust, hence the variable irrecoverable strain that builds up lateraly. During the coseismic phase, stress drops to a lower level, leaving persistent strain within the forearc.


Dima Grigoriev
email: dmitry.grigoryev@math.univ-lille1.fr

Tropical recurrent sequences

Keywords: tropical recurrent sequence, periodic sequence, tropical entropy

Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case there are periodic tropical recurrent sequences which are similar to classical linear recurrent sequences. A question is studied when there exists a non-periodic tropical recurrent sequence satisfying a given vector, and partial answers are provided to this question. Also an algorithm is designed which tests existence of non-periodic tropical recurrent sequences satisfying a given vector with integer coordinates. Finally, we introduce a tropical entropy of a vector, provide some bounds on it and extend this concept to tropical multivariable recurrent sequences.

Introduction

A classical (linear) recurrent sequence {z l } l∈Z (e. g. Fibonacci numbers) satisfies conditions 0≤i≤n a i z i+k = 0, k ∈ Z, a 0 = 0, a n = 0. It well known that the linear space of all such sequences has dimension n and can be explicitly described via the roots of polynomial 0≤i≤n a i x i and the derivatives in case of multiple roots.

We study tropical recurrent sequences satisfying similar tropical linear polynomials min 0≤i≤n {a i + y i+k }, k ∈ Z where as it is adopted in tropical algebra [START_REF] Maclagan | Introduction to Tropical Geometry[END_REF] we assume that the minimum is attained for at least two different indices 0 ≤ i 1 (k) < i 2 (k) ≤ n for each k ∈ Z. We say that in this case a tropical recurrent sequence y = {y i ∈ R ∪ {∞}} i∈Z satisfies vector a := (a 0 , . . . , a n ) ∈ (R ∪ {∞}) n+1 , a 0 < ∞, a n < ∞. One can treat a tropical recurrent sequence as a solution of an (infinite) tropical Macaulay matrix whose rows are obtained from vector a by all possible shifts. We mention that Macaulay matrix plays a key role in the tropical Nullstellensatz [START_REF] Grigoriev | On a tropical dual Nullstellensatz[END_REF], [START_REF] Grigoriev | Tropical effective primary and dual Nullstellensaetze[END_REF].

On the other hand, one can view tropical linear recurrence as a tropical analog of shift equations with constant coefficients (note that in the classical setting shift equations extend difference equations). Also one can obtain tropical recurrent sequences by means of tropicalizing linear recurrent sequences of the form 0≤i≤n A i Z i+k = 0, k ∈ Z over Puiseux series.

Throughout the paper (except for sections 5, 6) we impose the requirement of minimality of tropical recurrent sequences: for any j ∈ Z one can not diminish y j keeping all the rest y i , i = j without violation of being a tropical recurrent sequence satisfying a.

A description of tropical recurrent sequences satisfying a given vector a is more complicated than its classical counterpart, and we don't provide a complete answer.

The collection of points {(i, a i ) : 0 ≤ i ≤ n} ⊂ R 2 we call Newton graph of a. A crucial feature of a is its Newton polygon P (a) on the plane which is the convex hull of the vertical rays {(i, b ≥ a i ), 0 ≤ i ≤ n}. For each (bounded) edge of P (a) with a slope σ there exists a tropical recurrent sequence {y j } j∈Z satisfies a where points (j, y j ), j ∈ Z are located on a line with the slope -σ. There can be more general periodic tropical recurrent sequences, so for some period d ∈ Z it holds y j+d -y j = -σd, j ∈ Z.

In a certain (informal) sense periodic tropical recurrent sequences are similar to classical recurrent sequences. On the other hand, for some vectors a there exist non-periodic tropical recurrent sequences satisfying a. We study for which a they exist.

In section 1 vectors a are considered such that all the finite points (i, a i ), a i < ∞ from Newton graph lie on a single (bounded) edge of P (a). We show that all the tropical recurrent sequences satisfying a are periodic iff the points (i, a i ), a i < ∞ form an arithmetic progression with some difference d. In the latter case any tropical recurrent sequence satisfying a has period d.

In section 2 a more general situation is studied when P (a) has a single (bounded) edge. First, we note that if points (i, a i ) lying on this edge do not form an arithmetic progression then there exists a non-periodic tropical recurrent sequence satisfying a. When, on the contrary, these points form an arithmetic progression with the difference 2 we prove that all the tropical recurrent sequences satisfying a are periodic iff points (i, a i ) not lying on the edge, also form an arithmetic progression.

In section 3 vectors a are considered with an arithmetic progression having the difference 3 of points (i, a i ) lying on the (single) edge. We provide two examples of a: one having only periodic tropical recurrent sequences satisfying a, and another one with non-periodic sequences. These two examples demonstrate that the existence of non-periodic tropical recurrent sequences satisfying a can not be expressed just in terms of arithmetic progressions. It would be interesting to give an explicit answer to the question of existence of non-periodic tropical recurrent sequences satisfying a.

In section 4 we design an algorithm which tests for vector a = (a 0 , . . . , a n ) with integer coordinates a i ∈ Z, 0 ≤ i ≤ n whether there exists a non-periodic tropical recurrent sequence satisfying a.

In section 5 we introduce a tropical entropy H(a) and a tropical minimal entropy h(a), they fulfil inequalities 0 ≤ h(a) ≤ H(a). We provide an upper bound on H(a) and calculate h(a) and H(a) for some examples of vectors a.

In section 6 we extend the concepts of the tropical (respectively, minimal) entropy to tropical multivariable recurrent sequences. Again we provide an upper bound on H(a) and calculate H(a) for an example of vector a.

Tropical recurrent sequences extend recognizable sequences [START_REF] Berstel | Rational series and their languages[END_REF] z k = cB k e, 0 ≤ k ∈ Z where B is a square matrix and c, e are vectors over a tropical semiring. Observe that recognizable sequences fulfil a tropical linear recurrence due to a tropical version of Cayley-Hamilton theorem [START_REF] Akian | Linear independence over tropical semirings and beyond[END_REF]. Schützenberger's theorem (see [START_REF] Berstel | Rational series and their languages[END_REF]) characterizes recognizable sequences by means of rationality of the generating series 0≤k<∞ z k X k . Recognizable sequences are ultimately periodic (so, become periodic for sufficiently large k) (see e. g, [START_REF] Krob | A complete system of identities for one letter rational expressions with multiplicities in the tropical semiring[END_REF], [START_REF] Gaubert | Rational series over dioids and discrete event systems[END_REF]) which differs from tropical recurrent sequences. A related work is [START_REF] Cohen | Algebraic tools for the performance evaluation of discrete event systems[END_REF] where a subclass of non-decreasing recognizable sequences was introduced, and it plays a role in the modelling of timed discrete event systems [START_REF] Baccelli | Synchrolization and linearity[END_REF].

1 Tropical recurrent sequences satisfying a vector with Newton graph located on a line Definition 1.1 We say that a tropical recurrent sequence y = {y j } j∈Z satisfying vector a is minimal if for any j ∈ Z there exists k ∈ Z, j -n ≤ k ≤ j such that a j-k +y j = min 0≤i≤n {a i +y i+k }. In other words, one can not diminish y j keeping all the rest y l , l = j.

One can plot a i as point (i, a i ) on the plane, respectively, y j as (j, y j ). Remark 1.2 Making a suitable affine transformation of the plane one can assume w.l.o.g. that Newton polygon P (a) contains an edge situated on the abscissas axis. Hence {y i = 0 : i ∈ Z} is a tropical recurrent minimal sequence satisfying a.

In this section we study the case when all the points (i, a i ) (so, Newton graph of a) are located on a single bounded edge of Newton polygon P (a). Making a suitable affine transformation of the plane (cf. Remark 1.2) one can assume w.l.o.g. that this edge is situated on the abscissas axis, so either a i = 0 or a i = ∞ for all i ∈ Z. In particular, a 0 = a n = 0.

Note that if y (1) , y (2) are two tropical recurrent sequences satisfying a then min{b 1 + y (1) , b 2 + y (2) } is also the tropical recurrent sequence satisfying a, where b 1 , b 2 ∈ R.

There is always the trivial infinite tropical recurrent sequence {y j = ∞, j ∈ Z}, so we suppose that tropical recurrent sequences under consideration differ from the infinite one. Proposition 1.3 Let for vector a = (a 0 , . . . , a n ) hold a i = 0 for all finite a i . Then all the tropical recurrent minimal sequences satisfying a are periodic iff all i for which a i = 0 form an arithmetic progression. If d is the difference of this progression then every tropical recurrent minimal sequence satisfying a is periodic with the period d.

Proof. First assume that set S := {i ∈ Z : a i = 0} does not form an arithmetic progression. We claim that for any 0 < b ∈ R ∪ {∞} and k ∈ Z the tropical recurrent sequence y i = b for i -k ∈ S and y i = 0 otherwise, satisfies a. Suppose the contrary. To simplify notations assume w.l.o.g. that k = 0. Then there exists s ∈ Z such that minimum min 0≤i≤n {a i + y i+s } is attained once. When s = 0 we have a j + y j = b = min 0≤i≤n {a i + y i } for each j ∈ S. In particular, a 0 + y 0 = a n + a n = b, and we get a contradiction with that the minimum is attained once. When s > 0 we have 0 = a n + y n+s = min 0≤i≤n {a i + y i+s }. Since the minimum is attained once, we conclude that for each 0 ≤ i < n, i ∈ S it holds a i + y i+s = b, therefore S forms an arithmetic progression with the difference s, we get a contradiction. A similar argument works when s < 0, in this case 0 = a 0 + y s = min 0≤i≤n {a i + y i+s }. The claim is proved.

We now show that the sequence y is minimal (see Definition 1.1). Indeed, for each i ∈ S we have b = a i + y i = min 0≤l≤n {a l + y l }. On the other hand, for each i / ∈ S we have 0 = a 0 + y i = min 0≤l≤n {a l + y i+l } due to the proved above claim.

Before proving the other implication in Proposition 1.3 we make the following remark.

Remark 1.4 Thus, in case when S does not form an arithmetic progression one can modify the zero tropical recurrent sequence {y i = 0, i ∈ Z} increasing y i+k for i ∈ S by b k = b > 0. Moreover, one can take arbitrary integers . . . , k -1 , k 0 , k 1 , . . . such that k l+1 -k l > 2n, l ∈ Z, and for each k l modify the zero solution by b k l > 0 as described above. Thus, one obtains an uncountable number of modifications (just by choosing {k l } l∈Z regardless of b k l ) of the zero tropical recurrent sequence, satisfying a.

Let now S form an arithmetic progression with a difference d (i. e. S = {id : 0 ≤ i ≤ m}). Let a tropical recurrent minimal sequence y satisfy a. Fix 0 ≤ i 0 < d for the time being. To prove the last statement of Proposition 1.3 on periodicity it suffices to verify that the subsequence {y id+i 0 : i ∈ Z} of y is constant for i ∈ Z. Denote c := min 0≤i≤m {y id+i 0 } = min 0≤i≤m {a id + y id+i 0 }. Then this minimum is attained for two different 0 ≤ i 1 < i 2 ≤ n. Observe that for every i 1 < i < i 2 it holds y id+i 0 = c as well due to the minimality of y (see Definition 1.1). In addition, observe that c = min -1≤i≤m+1 {y id+i 0 }. Indeed, if on the contrary y (m+1)d+i 0 < c then minimum min 0≤i≤m {a id + y (i+1)d+i 0 } = y (m+1)d+i 0 is attained only once, which contradicts that y is a tropical recurrent sequence satisfying a. Similarly, one shows that y -d+i 0 ≥ c. Repeating this argument recursively one deduces that c = min -∞<i<∞ {y id+i 0 }. Considering c = a 0 + y i 2 d+i 0 = min 0≤i≤m {a id + y (i+i 2 )d+i 0 } one gets that there exists i 2 < i 3 ≤ i 2 + m such that y i 3 d+i 0 = c. Then as above one obtains that y id+i 0 = c for any i 2 < i < i 3 . Similarly, there exists i 1 -m ≤ i 4 < i 1 such that y i 4 d+i 0 = c. Hence y id+i 0 = c for any i 4 < i < i 1 . Repeating this argument one concludes that y id+i 0 = c for any i ∈ Z.

This completes the proof that y is periodic with the period d. 2

2 Tropical recurrent sequences for a Newton polygon with a single edge and period 2

In the previous section we studied the case when points (i, a i ), 0 ≤ i ≤ n from Newton graph are located on a line. Note that in general, Newton polygon P (a) has two unbounded edges and several bounded ones. In the present section we suppose that P (a) has a single bounded edge. Similar to the previous section, making a suitable affine transformation of the plane one can assume w.l.o.g. that this edge lies on the abscissas axis. Again as in the previous section we consider set S := {0 ≤ i ≤ n : a i = 0}. In particular, 0, n ∈ S.

Remark 2.1 First, consider the case when S does not form an arithmetic progression. Then similar to Remark 1.4 one can modify the zero tropical recurrent sequence y j = 0, j ∈ Z by replacing y j = b for j ∈ S, while b > 0 should be taken less than min{a i : a i > 0}. Thus, again one obtains an uncountable number of periodic tropical recurrent minimal sequences satisfying a.

So, from now on we assume that S forms an arithmetic progression with a difference d. In the present section we study the case d = 2 and investigate when there is a non-periodic tropical recurrent minimal sequence satisfying a. In particular, in this case n is even and 2i ∈ S for all 0 ≤ 2i ≤ n. Theorem 2.2 Let Newton polygon P (a) have a single bounded edge (assuming w.l.o.g. that this edge is located on the abscissas axis), and the points S from Newton graph of a on this edge form an arithmetic progression with the difference 2 starting with the point 0 and finishing with the point n. Then any tropical recurrent minimal sequence satisfying a is periodic iff all a i with odd i are equal. In the latter case any tropical recurrent minimal sequence satisfying a, has period 2.

Remark 2.3

In other words, under the conditions of the Theorem any tropical recurrent minimal sequence is periodic iff points (i, a i ) from Newton graph of a are located on two parallel lines: one for even 0 ≤ i ≤ n and the second for odd

1 ≤ i ≤ n -1.
Proof of the theorem. First consider the case when not all a i with odd i are equal. Denote c := min{a i : odd i} > 0, e := min{a i : odd i, a i > c} > c and C := {i : a i = c}. Take a periodic (with the period 2) tropical recurrent sequence y (0) := {y

i i ∈ Z}) putting • y 2i := e, 0 ≤ 2i ≤ n; • y 2i+1 := e -c, 2i + 1 ∈ C, 1 ≤ 2i + 1 ≤ n -1; • y 2i+1 = 0, 2i + 1 / ∈ C, 1 ≤ 2i + 1 ≤ n -1
, while keeping the rest of the coordinates unchanged.

Let us verify that y satisfies a. For any odd k ≥ 3 minimum 0 = a n +y n+k = a n-2 +y n-2+k = min 0≤i≤n {a i +y i+k } is attained at least twice. Similarly, for any odd k ≤ -3 minimum 0 = a 0 +y k = a 2 +y k+2 = min 0≤i≤n {a i +a i+k } is attained at least twice as well. For k = ±1 minimum min 0≤i≤n {a i +y i+k } = 0 is attained at least twice since there is an odd 1 ≤ i ≤ n -1 such that i / ∈ C. For k = 0 minimum min 0≤i≤n {a i + y i } = e is attained (for any even 0 ≤ i ≤ n, for any i ∈ C and for any odd i / ∈ C such that a i = e) at least twice as well. For an even k = 0 minimum min 0≤i≤n {a i + y i+k } = c. For an even k ≥ 4 this minimum is attained at least twice for c = a n +y n+k = a n-2 +y n-2+k = min 0≤i≤n {a i +y i+k }. Similarly, for an even k ≤ -4 this minimum is also attained at least twice for c = a 0 + y k = a 2 + y k+2 = min 0≤i≤n {a i + y i+k }. Consider the case k = 2 (the case k = -2 can be considered in a similar way). We have a n + y n+2 = 0 + c. We claim that there is an odd 1 ≤ 2i -1 ≤ n -1 such that a 2i-1 + y 2i+1 = c. Assume that there exists 2i -1 ∈ C, 1 ≤ 2i -1 ≤ n -3 for which 2i + 1 / ∈ C. Then a 2i-1 = c, y 2i+1 = 0, which proves the claim under the assumption. If such 2i -1 ∈ C does not exist then n -1 ∈ C, hence a n-1 + y n+1 = c + 0, which proves the claim.

One can check the minimality of y (see Definition 1.1) for any (odd) 2i + 1 such that y 2i+1 = 0 with the help of an appropriate even k. For an odd 2i + 1 ∈ C (in this case y 2i+1 = e -c) one uses k = 0. Also k = 0 is used for even 0 ≤ 2i ≤ n (in this case y 2i = e). For an even |2i| ≥ n + 2 an appropriate odd k is involved.

Actually, one can take any 0 < q < d -c and modify y (0) putting

• y 2i = c + q, 0 ≤ 2i ≤ n; • y 2i+1 = q, 2i + 1 ∈ C, 1 ≤ 2i + 1 ≤ n -1; • y 2i+1 = 0, 2i + 1 / ∈ C, 1 ≤ 2i + 1 ≤ n -1,
while keeping the rest of the coordinates unchanged.

Again as in the proof of Proposition 1.3 one can modify y (0) changing y k l , y k l +1 , . . . , y k l +n as described above for integers . . . < k -1 < k 0 < k 1 < . . . such that k l+1 -k l > 2n for all integers l. Thus, there is an uncountable number of non-periodic tropical recurrent minimal sequences satisfying a. Now we consider a such that a 2i = 0, 0 ≤ 2i ≤ n; a 2i+1 = c > 0, 1 ≤ 2i + 1 ≤ n -1 and prove that any tropical recurrent minimal sequence y satisfying a, is periodic with the period 2.

Denote b := min -n≤i≤n {y i }. Similar to the proof of Proposition 1.3 one can deduce that b = min -n-1≤i≤n+1 {y i } and further by recursion that b = min -∞<i<∞ {y i }. Let y i = b for some even -n ≤ i ≤ n (an odd i can be considered in a similar way). Denote B := {2i : y 2i = b}. For any pair of adjacent elements 2i 1 < 2i 2 of B one has 2(i 2 -i 1 ) ≤ n, because otherwise, minimum min 0≤i≤n {a i + y i+2i 1 } = b is attained only once for i = 0. Therefore, for every even k we have min 0≤i≤n {a i + y i+k } = b, and for every odd l we have min 0≤i≤n {a i + y i+l } ≤ b + c, in addition y l ≤ b + c due to the minimality of y (see Definition 1.1). Again due to the minimality y k = b for every even k.

Denote p := min{y l : -n ≤ l ≤ n, odd l} ≤ b + c. If p = b + c then one deduces that y l = b + c for any odd l arguing as above by recursion on |l|, thus y is periodic with the period 2. So, assume that p < b + c. Arguing as above we conclude that p = min{y l : odd l ∈ Z} and that for all odd integers . . . < l -1 < l 0 < l 1 < . . . for which y l j = p, we have l j+1 -l j ≤ n. Hence due to the minimality of y we get that y l = p for any odd l. Thus, y is periodic with the period 2. Theorem is proved. 2 3 Newton polygon with a single edge and period greater than 2

So far, we considered vector a such that its Newton polygon P (a) had a single bounded edge (recall that w.l.o.g. one can assume that this edge is situated on the abscissas axis). Moreover, one can suppose that the set S = {i : a i = 0} for the points from Newton graph on this edge forms an arithmetic progression with a difference d (otherwise, as we have shown above, there would be a nonperiodic tropical recurrent minimal sequence satisfying a). We have given a complete answer to the question of existence of non-periodic minimal sequences for d = 1, 2. In the present section we provide examples for d = 3 which demonstrate that the answer is more complicated in this case and depends not only on the properties to be arithmetic progressions as for d = 1, 2. It would be interesting to give a complete answer for d ≥ 3. Proof. First consider the case c > 2b (the case b > 2c can be studied in a similar way). Take the following periodic tropical recurrent minimal sequence satisfying a: y Similar to the proofs of Proposition 1.3 and Theorem 2.2 one can choose integers . . . < k -1 < k 0 < k 1 < . . . and reals 0 < e l ≤ c-2b such that k l+1 -k l ≥ 3, and modify y (0) putting y 3k l +1 = 2b + e l ; y 3k l +2 = b + e l ; y 3k l +4 = 2b + e l for all integers l. Thus, one achieves a non-countable number of non-periodic tropical recurrent minimal sequences satisfying a. Now we study a fulfilling inequalities b < c ≤ 2b (the case of inequalities c < b ≤ 2c is considered in a similar way). Let a tropical recurrent minimal sequence y satisfy a. Denote q := min -3≤i≤3 {y i }. Arguing as in the proofs of Proposition 1.3 and Theorem 2.2, we conclude that q = min -4≤i≤4 {y i }, and continuing this argument we get by induction that q = min -∞<i<∞ {y i }. One can assume w.l.o.g. that y 3i 0 = q for some integer i 0 . Since minimum min 0≤i≤3 {a i + y 3i 0 +i } is attained at least twice, we deduce that y 3i 0 +3 = q. Continuing in this way, we deduce that y 3j = q for every integer j. Now we consider coordinates for an arithmetic progression {y 3i+2 : i ∈ Z}. Denote r := min 0≤i≤1 {y 3i+2 }. Hence r ≤ q + b since minimum min 0≤i≤3 {a i + y i+2 } ≤ a 1 + y 3 = b + q < a 2 + q ≤ c + y 4 should be attained at least twice. Arguing as above, we deduce that r = min -∞<i<∞ {y 3i+2 }. If r < q + b then since min 0≤i≤3 {a i +y 3j+i+2 } ≤ a 0 +r = r < b+q = a 1 +y 3j+3 < c+q ≤ a 2 +y 3j+4 for any integer j, we conclude that y 3i+2 = r for every integer i. Now let r = q + b. Since min 0≤i≤3 {a i + y 3j+i+1 } ≤ a 2 + q = c + q ≤ 2b + q = b + r ≤ a 1 + y 3j+2 , we get that if y 3j+2 > r then a 1 + y 3j+2 does not attain minimum in min 0≤i≤3 {a i + y 3j+i+1 }, hence a 0 + y 3j+2 = y 3j+2 attains the minimum in min 0≤i≤3 {a i + y 3j+i+2 } ≤ a 1 + y 3j+3 = b + q = r according to the minimality of y at y 3j+2 (cf. Definition 1.1). Thus, y 3i+2 = r for every integer i.

Finally, we consider coordinates for an arithmetic progression {y 3i+1 : i ∈ Z}. Due to the minimality of y one deduces that y 3i+1 ≤ t 0 := min{b+r, c+q} for every integer i. Denote t := min 0≤i≤1 {y 3i+1 } ≤ t 0 . Arguing as above, we conclude that t = min -∞<i<∞ {y 3i+1 }. When t = t 0 , we have obviously, y 3i+1 = t for every integer i. When t < t 0 , arguing as above we also show that y 3i+2 = t for every integer i. This completes the proof of the Proposition. 2 Remark 3.2 When P (a) has several edges g 1 , . . . , g k with slopes σ 1 < • • • < σ k , respectively, then as a tropical recurrent sequence satisfying a one can take points {(i, y i ) : i ∈ Z} lying on the edges of an (infinite in both directions) convex polygon having edges g k , g k-1 , . . . , g 1 with the slopes -σ k , -σ k-1 , . . . , -σ 1 , respectively, such that g j is not shorter than g j , 1 ≤ j ≤ k.

Conversely, in section 4 [START_REF] Grigoriev | On a tropical dual Nullstellensatz[END_REF] it is proved, in fact, that if for each 0 ≤ i ≤ n point (i, a i ) lies on the boundary of P (a) then any tropical recurrent sequence satisfying a has the described form.

Algorithm testing existence of a nonperiodic tropical recurrent sequence

Let a = (a 0 , . . . , a n ) ∈ Z be a vector with integer coordinates whose Newton polygon has a single bounded edge which is located on the abscissas axis.

Definition 4.1 We say that a vector z = (z 0 , . . . , z N ) ∈ R N +1 satisfies a and is minimal if for each 0 ≤ k ≤ N -n minimum min 0≤i≤n {a i + z i+k } is attained at least twice and for each n ≤ j ≤ N -n there exists j -n ≤ k ≤ j such that a j-k + z j = min 0≤i≤n {a i + z k+i } (cf. the definition of a tropical recurrent sequence and Definition 1.1).

In this section we prove the following theorem. Theorem 4.2 There is an algorithm which for a vector a = (a 0 , . . . , a n ), 0 ≤ a i ≤ M, 0 ≤ i ≤ n, a 0 = a n = 0 tests whether there exists a non-periodic tropical recurrent minimal sequence satisfying a. The complexity of the algorithm does not exceed M O(n) .

Proof. It suffices to consider sequences y = {y i , i ∈ Z} with integer coordinates y i . Denote q := min -n≤i≤n {y i }. Replacing y i by y i -q, i ∈ Z one can assume w.l.o.g. that q = 0. Arguing as in the proofs of Proposition 1.3 and of Theorem 2.2 above, we conclude that min -∞<i<∞ {y i } = 0.

Denote S := {i ∈ Z : y i = 0}. Again arguing as in the proofs of Proposition 1.3 and of Theorem 2.2 above, we deduce that for any pair of adjacent element i 1 < i 2 of S inequality i 2 -i 1 ≤ n holds.

Proof. It suffices to show that if one of the connected components K of G = G(a) (to define connected components we consider G as an undirected graph) is not a simple cycle then there exists a non-periodic path in G (see Lemma 4.3). To show this, treat G as a partial order on its vertices. Then the vertices of G are partitioned into classes of equivalence (two vertices belong to the same class if there exists a cycle in G containing both vertices). The classes of equivalence are partially ordered (this order is induced by G).

Since G contains no vertices without in-coming or out-coming arrows, every minimal or maximal (with respect to the mentioned above partial order) class of equivalence contains a cycle with at least one arrow. Therefore, either there is (in K) a class of equivalence being not a simple cycle or there are at least two classes of equivalence. First, let there be a class of equivalence C in K being not a simple cycle. Then C contains suitable three vertices v 0 , v 1 , v 2 and two arrows (v 0 , v 1 ), (v 0 , v 2 ). There exists a non-periodic path passing through the vertices of C, and each time passing v 0 the path chooses either v 1 or v 2 as the next vertex in an arbitrary non-periodic manner.

Second, let there be at least two classes of equivalence in K. Then there exists a path in G of the form v

0 v 1 . . . v k (= v 0 )u 1 . . . u l v 0 v 1 . . . v q (= v 0 ) such that the cycle v 0 v 1 . . . v k (= v 0 )
belongs to one class of equivalence, while the cycle v 0 v 1 . . . v q (= v 0 ) belongs to another class of equivalence. As a non-periodic path in G one can take a repetition of an infinite number of times of cycle v 0 v 1 . . . v k (= v 0 ) (for negative moments of time till 0), subsequently u 1 . . . u l (for the moments 1, . . . l), and subsequently a repetition of an infinite number times of cycle v 0 v 1 . . . v q (= v 0 ) (starting with the moment l + 1). 2 Remark 4.5 (i) More generally, one can design an algorithm similar to Theorem 4.2 for vectors a = (a 0 , . . . , a n ) with rational coordinates a i ∈ Q, 0 ≤ i ≤ n looking for tropical recurrent minimal sequences of the form y = {y i /q : i ∈ Z, y i ∈ Z} where q being the common denominator of a 0 , . . . , a n ;

(ii) it would be interesting to design an algorithm similar to Theorem 4.2 for vectors a allowing real algebraic and infinite coordinates.

Tropical entropy

Let a = (a 0 , . . . , a n ) ∈ R n+1 . For 0 ≤ s ∈ Z denote by D s ⊂ R s (respectively, M s ⊂ R s ) the set of vectors satisfying a (respectively, satisfying in addition the minimality condition, see Definition ?? and section 4). Both D s and M s are polyhedral complexes [START_REF] Maclagan | Introduction to Tropical Geometry[END_REF]. Denote d s := dim D s , m s := dim M s .

When i + j = s denote by p : R s → R i , q : R s → R j the projection onto the first i coordinates and respectively, onto the last j coordinates. Since

p(D s ) ⊂ D i , q(D s ) ⊂ D j , p(M s ) ⊂ M i , q(M s ) ⊂ M j we have d i+j ≤ d i + d j , m i+j ≤ m i + m j .
Therefore, due to Fekete's subadditive lemma [START_REF] Schechter | Handbook of Analysis and its Foundations[END_REF] there exist limits (which coincide as well with the infimums of these sequences) Proof. The polyhedral complex D s is a union of polyhedra such that each of these polyhedra Q satisfies the following conditions. For every 0 ≤ j ≤ s-n there exists a pair 0 ≤ i 1 < i 2 ≤ n such that z j+i 1 + a i 1 = z j+i 2 + a i 2 = min 0≤i≤n {z j+i + a i } for any (z 1 , . . . , z s ) ∈ D s . For j = i 1 + 1 there exists a pair 0 ≤ i 3 < i 4 ≤ n fulfilling the similar conditions, hence (i 1 +1+i 3 )-(j +i 1 ) ≤ n. Therefore, there are at least s/n such pairs. Each such pair (j + i 1 , j + i 2 ) imposes a linear restriction

z j+i 1 -z j+i 2 = a i 2 -a i 1 on Q. Thus, d s ≤ s-s/n . 2
Example 5.2 Let now vector a be with a 0 = • • • = a n = 0. Consider a polyhedron consisting of vectors z = (z 1 , . . . , z s ) ∈ R s such that z (n+1)i+1 = z (n+1)i+2 = 0 for every i and the rest of the coordinates being arbitrary real nonnegative. Then z ∈ D s , hence d s ≥ s(1-2/(n+1)), thus, H(a) ≥ 1-2/(n+1). Now we prove an upper bound on H. Let y = (y 1 , . . . , y s ) ∈ D s . Suppose w.l.o.g. that min 1≤i≤s {y i } = 0. Denote I := {1 ≤ i ≤ s : y i = 0} and i 1 < i 2 < • • • being the consecutive elements of I. Observe that i j+2 -i j ≤ n for every j since otherwise, min 0≤l≤n {a l + y l+i j +1 } is attained only once for l = i j+1 -i j -1. Therefore, y belongs to a linear space {y i = 0, i ∈ I} with the dimension at most s(1 -2/(n + 1)) . Thus, H(a) = 1 -2/(n + 1).

Due to Proposition 1.3 any vector satisfying a and the condition of the minimality (see Definition 1.1) has all equal coordinates, so m s = 1, hence h(a) = 0. Remark 5.3 There is a gap between an upper bound on H from Proposition 5.1 and the latter example. The conjecture is that 1 -2/(n + 1) is an upper (sharp) bound on H.

We call a vector a regular if the set J := {i | a i < ∞} is an arithmetic progression and each point (i, a i ), i ∈ J is a vertex of the Newton polygon P (a). Theorem 5.4 If a vector a is not regular then H(a) ≥ 1/6. Corollary 5.7 A vector a is regular iff H(a) = 0. For non-regular a the inequality H(a) ≥ 1/6 holds.

Example 5.8 Now we give an example of a vector a = (a 0 , a 1 , a 2 , a 3 ) with a positive h(a). Put a 0 = a 3 = 0, a 2 > 2a 1 > 0 (see Proposition 3.1). Then at the beginning of the proof of Proposition 3.1 a family of vectors satisfying a is constructed for an arbitrary k 1 < k 2 < • • • with k i+1 -k i = 3 for all i. which demonstrates that h(a) ≥ 1/9.

On the contrary, if 2a 1 ≥ a 2 > a 1 > 0 then h(a) = 0 (cf. the proof of Proposition 3.1). Remark 5.9 (i) For a with a single bounded edge of its Newton polygon (cf. sections 1, 2, 3) is it true that h(a) = 0 iff all the tropical minimal recurrent sequences satisfying a are periodic?

(ii) For a vector a from Example 5.2 it was, actually, established that when (n + 1)|(s -r), 0 ≤ r ≤ n, function d s = (s -r)(n -1)/(n + 1) + r in case if 0 ≤ r < n and d s = (s -n)(n -1)/(n + 1) + n -1 if r = n. Is it true that for an arbitrary vector a function d s is linear with the leading coefficient H(a) for s from each fixed arithmetic progression with the difference n + 1?

6 Tropical multivariable recurrent sequences For a parallelepiped in the lattice Q ⊂ Z m with the sides q 1 , . . . , q m , respectively, we say that {z I : I ∈ Q} satisfies a if for any J ∈ Z m the minimum min I {a I + z I+J } is attained at least twice, provided that I + J ∈ Q for each I from the support of a. Clearly, one could consider an arbitrary subset of Z m rather than just a parallelepiped.

Similar to section 5 define d q 1 ,...,qm := d q 1 ,...,qm (a) to be the dimension of the tropical linear prevariety {z I : I ∈ Q} satisfying a (i. e. a set of points satisfying a finite family of tropical linear equations [START_REF] Maclagan | Introduction to Tropical Geometry[END_REF]). Analogously one defines m q 1 ,...,qm with respect to tropical multivariable minimal recurrent sequences. There exist limits H(a) := lim q 1 ,...,qm→∞ d q 1 ,...,qm /(q 1 • • • q m ), h(a) := lim q 1 ,...,qm→∞ m q 1 ,...,qm /(q 1 • • • q m ) which we also call the tropical entropy (respectively, the tropical minimal entropy) of a. Again 0 ≤ h(a) ≤ H(a). One can prove the following proposition similarly to the proof of Proposition 5.1.

  i ∈ Z} satisfying a. Let us modify it (denote the modified sequence by y := {y

Proposition 3 . 1

 31 Let vector a : a 0 = a 3 = 0, a 1 = b, a 2 = c; b, c > 0, b = c. Any tropical recurrent minimal sequence satisfying a is periodic iff either b < c ≤ 2b or c < b ≤ 2c. In this case any minimal sequence is periodic with the period 3.

  3i+2 := b; i ∈ Z. Consider a real 0 < e ≤ c -2b. We modify y (0) resulting in a non-periodic tropical recurrent minimal sequence y satisfying a: y 1 = 2b + e, y 2 = b + e, y 4 = 2b + e and keeping the rest of the coordinates of y (0) unchanged.

H

  = H(a) := lim s→∞ d s /s, h = h(a) := lim s→∞ m s /s which we call the tropical entropy of a (respectively, the tropical minimal entropy of a). Clearly, 0 ≤ h ≤ H. Proposition 5.1 H ≤ 1 -1/n

For a vector

  a = {a I ∈ R ∪ {∞} : I ∈ Z m } with a finite number of I such that a I ∈ R (the set of such I we call the support of a) we say that a tropical multivariable recurrent sequence {z I ∈ R : I ∈ Z m } satisfies a if for any J ∈ Z m the minimum min I {a I + z I+J } is attained at least for two different I. Similar to Definition 1.1 one can also define tropical multivariable recurrent minimal sequences.

Acknowledgements. The author is grateful to the grant RSF 16-11-10075, to MCCME for inspiring atmosphere, to A. Tiskin and to an anonymous referee for useful remarks.

For each j ∈ Z due to the minimality of y (see Definition 1.1) there exists an integer k, j -n ≤ k ≤ j such that a j-k + y j = min 0≤i≤n {a i + y k+i }. On the other hand, there is 0 ≤ l ≤ n such that y k+l = 0, hence y j ≤ a l . Thus, y j ≤ M, j ∈ Z.

A vector z = (z 0 , . . . , z N ) ∈ {0, . . . , M } N +1 we treat as a word in the alphabet {0, . . . , M }. We agree that a subword of a word consists of its consecutive letters. We correspond to a sequence y = . . . y -1 y 0 y 1 . . . the sequence Y = Y (y) in the alphabet W := {0, . . . , M } 2n+1 consisting of all subwords of y of the length 2n + 1, so that (y i+1 . . . y i+2n+1 ) ∈ W is the next letter in Y after (y i . . . y i+2n ) ∈ W for all i ∈ Z.

Construct a directed graph G 0 := G 0 (a) whose vertices are elements of W which satisfy a and being minimal (see Definition 4.1). There is an arrow in G 0 from a vertex (z 0 . . . z 2n ) ∈ W to a vertex (z 0 . . . z 2n ) ∈ W iff z i+1 = z i , 0 ≤ i ≤ 2n -1 (in particular, loops are admitted). In other words, vector z 0 . . . z 2n z 2n = z 0 z 0 . . . z 2n ∈ {0, . . . , M } 2n+2 satisfies a and is minimal. (ii) conversely, for any infinite in both directions path p in G 0 there is a unique tropical recurrent minimal sequence y satisfying a such that p = p(Y (y));

(iii) sequence y is periodic with a period d iff Y is periodic with the period d as well.

Proof. The only non-trivial statement is (ii). To yield y = . . . y -1 y 0 y 1 . . . take consecutively the first components (being integers between 0 and M ) of the letters (from W ) of sequence Y produced from p. Then Y = Y (y) and p = p(Y (y)). To prove minimality of y take its arbitrary letter y j and make use of that the word Y j := (y j-n . . . y j . . . y j+n ) ∈ W is a minimal vector satisfying a since Y j is a vertex of G 0 , and thus a j-k + y j = min 0≤i≤n {a i + y k+i } for some j -n ≤ k ≤ j (see Definition 4.1). The proof of that y satisfies a is similar (and even easier). 2

Observe that vertices of G 0 without in-coming or out-coming arrows do not appear in infinite in both directions paths, that is why we remove such vertices from G 0 and the resulting (directed) graph denote by G := G(a). Remark 1.2 implies non-emptyness of G.

Theorem 4.2 follows from the next lemma. Proof. First consider the case when at least three points of a lie on a (bounded) edge of P (a). Similar to the beginnings of sections 1, 2 making suitable affine transformations one can suppose w.l.o.g. that an edge containing at least three points of a lies on the abscissas axis. Consider the points of a located on this edge: {(i, 0) : i ∈ I} where |I| ≥ 3. One can assume w.l.o.g. that the greatest common divisor GCD(I) of the differences i 1 -i 2 of all the pairs of the elements i 1 , i 2 ∈ I of I equals 1. Otherwise, one can consider separately all GCD(I) arithmetic progressions with the difference GCD(I).

Pick any three elements of I not all with the same parity, say 0, 2i, j w.l.o.g. where i ≥ 1 and j being odd. Consider the following tropical recurrent sequence z satisfying a. For odd indices 2l + 1 we put z 2l+1 = 0. For even indices 2(2qi + t), q ∈ Z, 0 ≤ t < i we put z 2(2qi+t) = 0 and z 2((2q+1)i+t) we put arbitrarily non-negative. Taking finite fragments (z 0 , . . . , z N ) with growing N we conclude that H(a) ≥ 1/4 in this case. Now assume that no edge of P (a) contains a point of a other than two vertices of this edge. Take an edge of P (a) with the vertices (i, a i ), (j, a j ) with the maximal difference j -i > 0. Again one can suppose w.l.o.g. that these vertices are (0, 0), and (n, 0). There exists i ∈ J such that n does not divide i since a is not regular. Among such i pick i 0 for which c :=

Then z satisfies a. Taking finite fragments (z 0 , . . . , z N ) with growing N we conclude that H(a) ≥ 1/4 for even n/k and H(a) ≥ k n/(2k) 2n

≥ 1/6 (the latter inequality becomes an equality when n/k = 3). 2

Example 5.5 Let vector a := (0, 1, 0). Then for any sequence (z 0 , . . . , z N ) satisfying a denote c := min{z 0 , . . . , z N }. Let z i = c for some 0 ≤ i ≤ N . For definiteness assume that i is even (the case of an odd i is considered in a similar way). Then z l = c for any even 0 ≤ l ≤ N . If m := min 0<2j+1≤N {z 2j+1 } < c+1 then z 2j+1 = m for any 0

Therefore, the number of odd 0 < 2j + 1 ≤ N for which z 2j+1 > c + 1 does not exceed N/4 , thereby H(a) ≤ 1/4. On the other hand, from the proof of Theorem 5.4 we conclude that H(a) ≥ 1/4 (in the notations of the proof of Theorem 5.4 n = 2, i 0 = 1). Remark 5.6 If all the points (i, a i ), a i < ∞, 0 ≤ i ≤ n are the vertices of the Newton polygon P (a) (thus, a being regular) then section 4 [START_REF] Grigoriev | On a tropical dual Nullstellensatz[END_REF] implies that H(a) = 0 (cf. also Remark 3.2). Our conjecture is that a stronger than in Theorem 5.4 bound H(a) ≥ 1/4 holds for not regular vector a. Proposition 6.1 Let the support of a be located in a cube with the side r. Then H(a) ≤ 1 -1/r m . Example 6.2 Let a be a vector with a I = 0 for all I = (i 1 , . . . , i m ), 1 ≤ i 1 , . . . , i m ≤ r, so its support is a cube with the side r -1. Then the following z = {z I } is a tropical multivariable recurrent sequence satisfying a. Put z l 1 r,l 2 r,...,lmr = z l 1 r+1,l 2 r,...,lmr = 0 for all integers l 1 , . . . , l m , and the rest of the coordinates of z being arbitrary non-negative. Therefore, H(a) ≥ 1 -2/r m .