Daniel A East

Duncan Sousa

Stephen R Martin

Thomas A Edwards

William Lehman

Daniel P Mulvihill
email: d.p.mulvihill@kent.ac.uk

Altering the stability of the Cdc8 overlap region modulates the ability of this tropomyosin to bind cooperatively to actin and regulate myosin

Keywords: Schizosaccharomyces pombe, Tropomyosin, Cdc8, acetylation, coiled-coil, fission yeast

is an evolutionarily conserved α-helical coiled-coil protein, dimers of which form end-to-end polymers capable of associating with and stabilising actin-filaments and regulate myosin function. The fission yeast, Schizosaccharomyces pombe, possesses a single essential Tm, Cdc8, which can be acetylated on its amino terminal methionine to increase its affinity for actin and enhance its ability to regulate myosin function. We have designed and generated a number of novel Cdc8 mutant proteins with amino terminal substitutions to explore how stability of the Cdc8-polymer overlap region affects the regulatory function of this Tm. By correlating the stability of each protein, its propensity to form stable polymers, its ability to associate with actin and to regulate myosin, we have shown the stability of the amino terminal of the Cdc8 α-helix is crucial for Tm function. In addition we have identified a novel Cdc8 mutant with increased amino-terminal stability, dimers of which are capable of forming Tm-polymers significantly longer than the wild-type protein. This protein had a reduced affinity for actin with respect to wild type, and was unable to regulate actomyosin interactions. The data presented here are consistent with acetylation providing a mechanism for modulating the formation and stability of Cdc8 polymers within the fission yeast cell. The data also provide evidence for a mechanism in which Tm dimers form end-to-end polymers on the actin-filament, consistent with a cooperative model for Tm binding to actin.

Introduction

Tropomyosin (Tm) is an evolutionarily conserved dimeric α-helical coiled coil protein that associates with actin filaments. Tm dimers associate on actin, in an end-to-end fashion to create an unbroken polymer that stabilises actin filaments, as well as regulate actomyosin interactions and actin filament dynamics in both muscle and non-muscle cells. For smooth and skeletal muscle tropomyosin, acetylation of the amino terminus is required for strong binding to actin [START_REF] Urbancikova | Requirement of amino-terminal modification for striated muscle alpha-tropomyosin function[END_REF][START_REF] Coulton | Functional homodimers and heterodimers of recombinant smooth muscle tropomyosin[END_REF] and thus to regulate myosin interaction. Acetylation allows the amino terminus of Tm to adopt a fully coiled-coil conformation [START_REF] Brown | Deciphering the design of the tropomyosin molecule[END_REF][START_REF] Frye | Structure of the tropomyosin overlap complex from chicken smooth muscle: insight into the diversity of N-terminal recognition[END_REF][START_REF] Greenfield | The effect of N-terminal acetylation on the structure of an N-terminal tropomyosin peptide and alpha alpha-tropomyosin[END_REF] and 'slot' into splayed carboxyl terminus of another Tm dimer [START_REF] Frye | Structure of the tropomyosin overlap complex from chicken smooth muscle: insight into the diversity of N-terminal recognition[END_REF][START_REF] Greenfield | Solution NMR structure of the junction between tropomyosin molecules: implications for actin binding and regulation[END_REF] forming a strong end-to-end interaction. The first 2-3 amino terminal residues of unacetylated Tms, such as those recombinantly expressed in bacteria, are non-helical and as such form weaker end-to-end interactions and have reduced affinity for actin.

Dimeric α-helical coiled coil proteins have a heptad pseudo-repeating sequence periodicity (residue positions are termed a to g -Figure 1A). Lateral interactions between residues of adjacent proteins (hydrophobic interactions between residues in the a and d positions and ionic interactions between residues in e and g positions) stabilise the coiled-coil structure [START_REF] Brown | Deciphering the design of the tropomyosin molecule[END_REF]7].

The mechanism by which Tm associates with actin is the subject of an ongoing debate. One model suggests the existence of specific actin interacting zones along the Tm filament which together provide sufficient electrostatic attraction with the actin polymer to promote binding [START_REF] Hitchcock-Degregori | Functions of tropomyosin's periodic repeats[END_REF][START_REF] Hitchcock-Degregori | Tropomyosin has discrete actin-binding sites with sevenfold and fourteenfold periodicities[END_REF]. A second model argues the intrinsic curvature of the Tm dimer facilitates its interaction with the actin filament in a cooperative mechanism, recently termed the 'Gestalt' model of Tm binding (for review see [START_REF] Holmes | Gestalt-binding of tropomyosin to actin filaments[END_REF]). Due to the weak affinity of individual Tm dimers for actin (> 20 µM), Tm must associate end-to-end to form a continuous polymer along the actin filament. The amino and carboxyl termini are essential for end-to-end interaction and therefore actin affinity [11]. Studies with endogenous and recombinant Tms have established that modifications to the amino and carboxyl termini alter end-toend interactions, actin affinity [START_REF] Urbancikova | Requirement of amino-terminal modification for striated muscle alpha-tropomyosin function[END_REF]12] and interaction with myosin [START_REF] Moraczewska | The ends of tropomyosin are major determinants of actin affinity and myosin subfragment 1-induced binding to F-actin in the open state[END_REF] along the actin filament .

The fission yeast Schizosaccharomyces pombe is a model organism amendable to cell, molecular, biochemical and genetical approaches. It has a single tropomyosin encoded by the cdc8 + gene which is essential for cell viability [START_REF] Balasubramanian | A new tropomyosin essential for cytokinesis in the fission yeast S. pombe[END_REF]. Previous work has shown that like muscle Tm, acetylation of Cdc8 is required for strong actin binding and myosin regulation [15] but is not essential for the viability of the fission yeast cell [START_REF] Coulton | The recruitment of acetylated and tropomyosin to distinct actin polymers permits the discrete regulation of specific myosins in fission yeast[END_REF].

In this study we use a series of novel Cdc8 amino terminal mutants to explore how stability of the Cdc8 overlap region affects the physical properties and regulatory function of this Tm. By correlating the stability of each protein and its propensity to form polymers with its ability to associate with actin and regulate myosin, it has been possible to establish that the stability of the Tmpolymer is crucial for its function. These data suggest a mechanism by which acetylation regulates the length of the Tm-polymers to permit efficient association with actin-filaments and thereby regulate myosin, consistent with a cooperative binding model of Tm binding to actin. A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

EXPERIMENTAL Yeast cell culture and strains

The yeast strains used in the study are wt (DMY244) h + his2-d1 ura4.d18 leu1.32; cdc8-110 (DMY333) h + cdc8-110 his2-d1 ura4.d18 leu1.32. Cell culture and maintenance were carried out according to [START_REF] Moreno | Molecular genetic analysis of fission yeast Schizosaccharomyces pombe[END_REF] using minimal medium (EMM2) supplemented with appropriate amino acids.

Molecular Modelling

The model presented in Figure 1C & D was generated using the following procedure: The S. pombe Cdc8 and chicken Tm alpha1 were aligned such that the N and C termini were each aligned according to the heptad repeats. The structure of the tropomyosin overlap complex from chicken smooth muscle (PDB code 3MTU [START_REF] Frye | Structure of the tropomyosin overlap complex from chicken smooth muscle: insight into the diversity of N-terminal recognition[END_REF]) was then mutated to the Cdc8 sequence using this alignment. Figure 1C & D was generated using PyMol (The PyMOL Molecular Graphics System, Version 1.3, Schrödinger, LLC).

Molecular Biology and Protein Purification

cdc8 mutants were generated by site directed mutagenesis using pJC20cdc8 + template [15] with appropriate oligonucleotides (Table 1). Each mutant was sequenced and also cloned into the fission yeast plasmid pREP41. Cdc8 proteins were purified as described previously [15] and concentrations were determined using a 280nm extinction coefficient of 2980cm -1 .

ATPase assays

Reactions were prepared in ATPase buffer (10 mM MOPS, 30mM NaCl, 5 mM MgCl 2 pH 7) containing 150 µM phosphoenolpyruvate (PEP), 150 µM NADH, 150 µM ATP, 10µl of pyruvate kinase/lactate dehydrogenase enzyme mix (Sigma-Aldrich). Varying concentrations of rabbit myosin II sub-fragment 1 (S1) were then added to the 1ml plastic cuvettes (Sarstedt) and allowed to equilibrate for 1 minute at 25 ºC. Actin or actin saturated with Cdc8 proteins (30 minute pre-incubations of actin and Cdc8 were performed at 25 ºC) was added to the reactions and mixed with a plastic stir-rod. Total reaction volume was 500 µl. Absorbance at 340 nm was monitored using a UV/Visible spectrophotometer (Varian Carey 50 Bio) connected to a PC and rates calculated from the slope of the curve.

Cosedimentation and quantitative electrophoresis

Cosedimentation assays were performed at 25 ºC by mixing 10 µM actin with increasing concentrations of Cdc8 in cosedimentation buffer (20 mM MOPS, 100 mM KCl, 5 mM MgCl 2 , pH 7.0) in an total of 100 µl. Reactions were incubated for 30 mins. Actin along with any bound tropomyosin was pelleted by centrifugation at 100,000 RPM for 20 minutes (Beckman Instruments TLA-100.1). Equal samples of pellet and supernatant were separated by SDS-PAGE. Gels were scanned using an Epson perfection 1640SU scanner with a transparency adaptor attached to a PC and analysis was performed using Scion Image software (Scion Corp., Frederick, MD). Viscosity analysis: A Cannon-Manning semi-microviscometer was used to determine the viscosity of 20 µM Cdc8 samples at 20 ± 1 ºC in 1 ml of viscometry buffer (20 mM MOPS, 5 mM MgCl 2 pH 7.0). NaCl concentration was increased from 0 to 250 mM. Kinematic viscosity was calculated using the manufacturer's predetermined microviscometer kinematic viscosity

Circular Dichroism

Circular dichroism measurements were made in 1 mm quartz cuvettes using a Jasco 715 spectropolarimeter. Cdc8 proteins were diluted in CD buffer (10 mM Potassium phosphate, 500 mM NaCl, 5 mM MgCl 2 pH 7.0) to a concentration of 0.4 mg/ml. Thermal unfolding data were obtained by monitoring the CD signal at 222 nm with a heating rate of 1 deg/min. CD data are presented as differential absorption (ΔA) or mean residue CD extinction coefficient (Δε mrw ) CD data were analyzed as a simple two-state equilibrium between the folded dimer (F 2 ) and the unfolded monomer (U).

The unfolding constant is defined as K = [U] 2 /[F 2 ] and the total concentration of protein (P 0 ) is given by P

0 = [U] + 2[F 2 ]
. The fraction of unfolded monomer present can be defined as F U = [U]/P 0 . Therefore:

(

F U at the mid-point of the unfolding transition (T m ) is equal to 0.5 by definition.

Therefore:

K Tm = P 0 and ∆G Tm = ∆H Tm -T m ∆S Tm = -RT m Ln(P 0 ). If one assumes that ∆C p ~ 0 then for all temperatures one has ∆H T = ∆ H Tm and ∆ S T = ∆ H Tm /T m + RLn(P 0 ).

Therefore:

(

CD data collected as a function of temperature were analyzed using the following equation:

CD T = (a+bT)(1-F u ) + (c+dT)F u
Where a,c and b,d are the intercepts and slopes of the of the pre-(a,b) and post-transition (c,d) regions and F u is obtained from equation (1) with K T from equation [START_REF] Coulton | Functional homodimers and heterodimers of recombinant smooth muscle tropomyosin[END_REF].

EM rotary shadowing:

Tm samples were diluted to 1.0 µM in a solution consisting of 5 mM Tris (pH 7.0), 5 mM KCl, 5 mM MgCl 2 and including 30 % glycerol were sprayed onto freshly cleaved mica. Platinum evaporation and rotary shadowing was carried out as in [START_REF] Sousa | Electron microscopy and persistence length analysis of semi-rigid smooth muscle tropomyosin strands[END_REF]. Electron microscopy was carried out on the shadowed molecules using a Philips CM120 electron microscope (FEI, Hillsboro, OR) at 120 kV and images digitized at 28,000 times magnification with a 2Kx2K F224HD slow scan CCD camera (TVIPS, Gauting, Germany). The recorded images of tropomyosin were then skeletonized following manual selection of 0.5 x 0.5 nm points every 4 to 5 nm along the center of the protein's longitudinal axis [START_REF] Li | The shape and flexibility of tropomyosin coiled coils: implications for actin filament assembly and regulation[END_REF][START_REF] Li | The relationship between curvature, flexibility and persistence length in the tropomyosin coiled-coil[END_REF]. The persistence length, ξ, was calculated, using the tangent angle correlation method, after determining θ (the deviation angles along tropomyosin molecules and polymers from an idealized straight rod) for segment lengths between 50 and 400 nm. Algorithms specifically tailored to determine ξ and θ were developed previously [START_REF] Li | The shape and flexibility of tropomyosin coiled coils: implications for actin filament assembly and regulation[END_REF][START_REF] Li | The relationship between curvature, flexibility and persistence length in the tropomyosin coiled-coil[END_REF]. Plots relating the inverse slope of <ln(cos θ)> to the segment length yields the persistence length values, where the factor of two in

F 2 2U K F 2 2U K Biochemical Journal
î s/ e s (è 2 )) ( cos ! =
accounts for the two dimensionality of the images [START_REF] Li | The shape and flexibility of tropomyosin coiled coils: implications for actin filament assembly and regulation[END_REF][START_REF] Li | The relationship between curvature, flexibility and persistence length in the tropomyosin coiled-coil[END_REF].

RESULTS

We wished to examine the role Cdc8 end-to-end interactions played in regulating the ability of this Tm to associate with actin and regulate myosin activity. To do this we designed mutations within the Cdc8 molecule that have the potential to disrupt the overlap region without affecting the integrity of the Cdc8 dimer. The carboxyl termini of different Tms diverge significantly, making it not only difficult to identify residues that may be key in proteinprotein interactions, but also challenging for correlating findings with Tms from other cell types. In contrast the amino termini of Tm proteins are highly conserved between yeast and metazoan species (Figure 1B) and it has been shown that small changes to this region can have a significant impact upon the function of this protein [START_REF] Monteiro | Functional alpha-tropomyosin produced in Escherichia coli. A dipeptide extension can substitute the amino-terminal acetyl group[END_REF].

Using homology alignments (Figure 1B) and molecular modelling of the Cdc8 overlap region (Figure 1C) we attempted to identify charged residues within the amino terminal region that have the potential to form electrostatic interactions with actin or the carboxyl terminus of an adjacent Cdc8 dimer, but were unlikely to affect dimer stability. Using this method we identified 2 conserved acidic residues (D2 & E6) within the Cdc8 amino terminus that sit in the b and f heptad positions of the coiled-coil (Figure 1A) and are therefore unlikely to disrupt the coiled-coil dimer. We also mutated a conserved acidic residue in a comparable b position (D16) to examine the effect charged residues predicted to lie outside of the overlap region had upon dimer stability or end-to-end interactions, as well interactions between the Tm and actin. Sequence alignment also revealed a short three amino acid motif of L-K-L (residues 11-13) present in each of the Tms except Cdc8. Although only half of this motif is predicted to lie within the overlap region, modelling predicted replacing the alanine at position 11 with a leucine residue (within the overlap) would fill a hydrophobic pocket present in the wild type dimer (Figure 1D). We were therefore intrigued to explore whether this divergence in amino terminal Tm sequence could play a role in modulating end-to-end interactions within the Cdc8 polymer.

A series of Cdc8 amino-terminal mutants were therefore generated in which either single acidic residues had been replaced (Cdc8-D2A, Cdc8-E6A, Cdc8-E6K & Cdc8-D16A) or the Ala-Arg-Ala residues at positions 11-13 had been replaced with the Leu-Lys-Leu motif, which is conserved in other Tm molecules (Cdc8-LKL). Each mutant protein was purified in a non-acetylated state from E. coli. For comparison wild type Cdc8 was purified in both acetylated and unacetylated forms from S. pombe and E. coli respectively. Each Cdc8 protein was subjected to SDS-PAGE (Figure 1E), electron spray mass-spectroscopy and spectrophotometric analyses to determine purity and concentration. The relative molecular mass of each full-length protein was as predicted, except for the D2A mutant, in which the amino-terminal methionine had been cleaved within the bacterial cell. This protein was subsequently named Cdc8-D2A* to highlight this amino terminal cleavage. To facilitate functional analyses of each mutant protein, each mutant gene was expressed within the fission yeast cell from the pREP41 expression vector and while none were found to have a dominant negative effect up the growth of the S. pombe cell (data not shown), only the cdc8-D2A gene was unable to complement the temperature sensitive growth defect of cdc8-110 cells while A c c e p t e d M a n u s c r i p t cells expressing cdc8-D16A displayed reduced growth (Figure 1F).

Circular dichroism (CD) analyses were performed upon each protein to assess the impact the amino acid substitutions had on the thermal stability of the dimer. The broad negative CD spectra peaks at 208 nm and 222nm consistent for α-helical structures were observed for the Cdc8 protein (Figure 2A). Spectra were equivalent for all Cdc8 proteins (not shown), and raw data curves for absorbance at 222 nm of three Cdc8 proteins highlighting the most dramatic effects are shown in Figure 2B. Figure 2C illustrates the first derivative plots for these data, from which midpoint melting temperatures (T m ) were calculated (Table 2). Acetylation of the wild type Cdc8 protein had a minimal effect upon the thermal stability of the Cdc8 dimer, increasing the T m only slightly from 34.7ºC to 35.4ºC (Table 2). For Cdc8-D2A* a significant decrease in thermal stability was observed, with the T m for this protein equal to 32.4ºC, which is likely to be more of a reflection of the lack of methionine 1 than loss of charge at the second residue. The E6A mutation had little effect on thermal stability, when compared to the WT protein, with a T m of 34.4ºC. The E6K mutation increased thermal stability to equal that of the wild-type acetylated Cdc8, 35.4ºC and the D16A mutation increased the T m slightly further to 35.9ºC. However, a dramatic increase in thermal stability was observed for the Cdc8-LKL protein (T m = 37.3ºC). These data confirm that as predicted (apart from the cleaved D2A), none of the amino terminal mutations had a detrimental effect on Cdc8 dimer stability.

End-to-end interactions between the amino and carboxyl termini of Cdc8 dimers allow the protein to polymerise. Viscometry was used to investigate the effect amino terminal mutations had upon the ability of Cdc8 to form filaments. In this assay a higher viscosity correlates with a greater capacity to form filaments. Figure 3A shows viscosity data for four Cdc8 proteins emphasizing the most dramatic effects, while a summary of viscosity for each of the Cdc8 proteins in the absence of NaCl is presented in Table 1. In the absence of NaCl Cdc8 has a viscosity of 1.08 centistokes (cSt). Acetlyation of Cdc8 had a significant effect increasing viscosity to 1.15 cSt. In contrast the Cdc8-D2A* protein was unable to form any detectable end-to-end interactions, as the viscosity of 1.05 this protein was equal to the buffer control containing no protein. The E6A, E6K and D16A mutations increased the viscosity to a value comparable with that of Cdc8-ACE, the recorded values were 1.13 cSt, 1.15 cSt and 1.14 cSt respectively. The most striking effect was observed with Cdc8-LKL, which had an increased viscosity of 1.22 cSt. In addition the viscosity of Cdc8-LKL in the presence of 200 mM NaCl was still greater than the buffer control. At this salt concentration the viscosity of each of the other samples was equal to that of the buffer. These data suggest that each amino terminal mutation had a significant impact upon endto-end interactions between the Cdc8 dimers, and that the D2A, E6A, E6K and D16A effects are electrostatically driven whereas the LKL mutant effects are more likely a hydrophobic effect.

Electron microscopy of rotary shadowed Tm was used to directly observe and compare the lengths of unacetylated Cdc8, Cdc8-ACE and Cdc8-LKL polymers in low salt conditions (5 mM). Figure 3B shows electron micrographs representative of what was typically observed for each protein. Non-acetylated Cdc8 protein was observed mainly as a population of single dimers and occasionally filaments of 2 interacting dimers, whereas the acetylated Cdc8 protein was seen as distinctly longer filaments, made up of two to four dimers (Figure 3C). Consistent with its high viscosity the Cdc8-LKL protein was capable of forming longer filaments, with a longer average length (Figure 3C). These observations were consistent with the viscosity data and gave confidence that the mutant proteins (except Cdc8-D2A*) form longer filaments than the unacetylated wild type protein.

From these images we were also able to determine the persistence length for each protein. The persistence length of unacetylated Cdc8 was calculated as 50 nm, which is approximately twice the length of a single Cdc8 dimer (24 nm). The persistence length of Cdc8-LKL was equal (90 nm, ≈ 4 dimers) to that of the acetylated protein. Therefore although the Cdc8-LKL protein is capable of forming longer filaments than the acetylated Cdc8 protein it does not stiffen the protein or inter-dimer interactions.

In addition to effects on Cdc8 stability and end-to-end contacts we wished to explore the effects of these mutations on other protein-protein interactions, specifically the interaction with actin. To do this we initially performed co-sedimentations experiments to determine the affinity of each protein for actin. Examples of SDS-PAGE gels used to determine binding affinity of three Cdc8 mutants are presented in Figure 4. In Figures 4A and4B the top bands are actin and the density remain approximately constant. The bottom bands are Cdc8 protein and the density increases as Cdc8 dimer concentrations increase from left to right. Figure 4A is an example of a Cdc8 mutant binding weakly to actin. Even at 20 µM Cdc8, only a small proportion of the protein is observed in the pellet fraction, while a large amount of Cdc8 remains in the supernatant. Conversely Figure 4B shows an example of a Cdc8 mutant which binds strongly to actin. A faint band can be observed in the pellet with as little as 0.2 µM Cdc8, a much denser band is visible at 8 µM Cdc8. Less Cdc8 can be observed in the supernatant at equivalent Tm concentrations. Typical binding curves for WT and Cdc8 mutants are presented in Figure 4D. Unacetylated Cdc8 has a K 50% of 2.76 µM, the acetylated protein has an increased affinity for actin with a K 50% of 0.46 µM. These observations for WT unacetylated and acetylated Tms were consistent with previously published data [15].

The truncated Cdc8-D2A* protein bound extremely weakly to actin and a binding curve could not be generated as the binding coefficient was estimated to be greater than 20 µM. The effect of the E6A and E6K mutations was to increase the affinity of these mutant proteins for actin, with values for K 50% of 0.32 µM and 0.45 µM respectively. These values are comparable with the WT acetylated protein. The K 50% for Cdc8-D16A is 2.27, and although lower, statistical analysis indicate no significant difference from the unacetylated wild type protein. This was expected as it is consistent with the mutation having no effect upon the Tm overlap region and also having minimal impact upon the interaction of the Tm dimer with actin. Surprisingly the Cdc8-LKL protein was observed in the pellet in the absence of actin (Figure 4C). As a result of this a binding curve could not be generated. Several attempts to determine a value using high salt conditions and lower centrifugation speeds were also unsuccessful and it was not possible to determine the actin affinity of this protein using this method. This is potentially either a consequence of an alteration to the overlap region or a change in shape of the Cdc8-LKL filament. Each K 50% was determined as an average 2.

Both unmodified and acetylated Cdc8 have been shown to be capable of regulating the interaction between myosins and actin [15], with the modified protein doing so significantly more effectively and this modification is necessary for the regulation of myosin II activity [START_REF] Coulton | The recruitment of acetylated and tropomyosin to distinct actin polymers permits the discrete regulation of specific myosins in fission yeast[END_REF][START_REF] Stark | Tropomyosin and myosin-II cellular levels promote actomyosin ring assembly in fission yeast[END_REF]23]. We therefore decided to use an ATPase assay to investigate the influence of Cdc8 amino terminal mutations on regulation of myosin ATPase activity, as this provides a measure of how the altered Cdc8 protein affects interactions outside of the Tm polymer. Figure 5A shows ADP release at increasing concentrations of S1 for WT and Cdc8 mutants. The rates of ADP release per micromole of S1 were expressed as a fraction of the maximal rate observed for actin and S1 in the absence of Tm protein (Table 2). Consistent with previous in vivo and in vitro studies [15, [START_REF] Coulton | The recruitment of acetylated and tropomyosin to distinct actin polymers permits the discrete regulation of specific myosins in fission yeast[END_REF][START_REF] Stark | Tropomyosin and myosin-II cellular levels promote actomyosin ring assembly in fission yeast[END_REF] both unacetylated and acetylated Cdc8 reduced the rate of actin activated ATPase activity of rabbit skeletal S1 to 0.46 and 0.17 of maximal.

When Cdc8-E6A or Cdc8-E6K were bound to actin, ATPase rates were reduced to 0.14 and 0.15 of maximal respectively, comparable to those observed for acetylated Cdc8. The Cdc8-D2A protein was not assayed owing to its extremely low affinity for actin. Cdc8-LKL had only a small inhibitory effect on S1 ATPase rate, reducing it to only 0.90 of maximal, however as an inhibitory effect was observed the affinity of Cdc8-LKL for actin was determined using this assay. ATPase rates were recorded in the presence of increasing Cdc8-LKL concentrations and a K 50% value of 5.2 µM calculated as the Cdc8-LKL concentration producing half of maximum inhibition. As a control the same method was used to calculate the K 50% of Cdc8 ACE . A value of 0.41 µM was determined, comparable to the value obtained using cosedimentation.

DISCUSSION

We have generated novel Cdc8 amino terminal mutants that were predicted to disrupt protein-protein interactions without adversely affecting the integrity of the Cdc8 dimer. A model of the Cdc8 overlap region was generated, based upon the structure of mammalian muscle Tm, not the yeast protein, and while each of the Tm overlap structures determined to date are based upon artificial constructs, they were predicted to mimic the endogenous structure [START_REF] Frye | Structure of the tropomyosin overlap complex from chicken smooth muscle: insight into the diversity of N-terminal recognition[END_REF][START_REF] Greenfield | Solution NMR structure of the junction between tropomyosin molecules: implications for actin binding and regulation[END_REF][START_REF] Murakami | Structural basis for tropomyosin overlap in thin (actin) filaments and the generation of a molecular swivel by troponin-T[END_REF].

The Cdc8 model predicted that none of the mutations generated in this study would have an adverse effect on Cdc8 dimer formation (Figure 1C). This is consistent with the prediction that the major effect amino-terminal acetylation has upon Tm is to stabilise the α-helical conformation at the amino terminus through the formation of intra-helical hydrogen bond to the amide hydrogen of the conserved hydrophobic residue in position 4 [START_REF] Brown | Deciphering the design of the tropomyosin molecule[END_REF][START_REF] Frye | Structure of the tropomyosin overlap complex from chicken smooth muscle: insight into the diversity of N-terminal recognition[END_REF]. The T m of Cdc8 was increased by less than a degree when the protein was acetylated, however the enthalpy of unfolding rose by 27 kJ mol -1 . This 15.8% change in enthalpy indicates that acetylation affects the stability of a large proportion of the protein, significantly more than the predicted overlap region alone. Comparison between modelled structures of the wild type and LKL mutant proteins suggest the side chains of leucine 11 in the mutant protein fills a hydrophobic pocket present in the wild type (Figure 1D), which is likely to have a significant positive effect upon the stability of the overlap region. Consistent with this the Cdc8-LKL protein had an 18.6% increase in enthalpy over the unacetylated protein, which corresponds to a stabilising effect on an even larger proportion of the dimer than the acetylated wild type protein.

The propensity of Cdc8 to form filaments as measured by viscosity was affected by amino terminal modifications. Acetylation of Cdc8 increased its viscosity significantly when compared to the unacetylated protein. This is consistent with the stabilising effect acetylation is predicted to have on the amino terminal α-helix, through the formation of intra-helical hydrogen bonds with the conserved hydrophobic residue four amino acids away [START_REF] Frye | Structure of the tropomyosin overlap complex from chicken smooth muscle: insight into the diversity of N-terminal recognition[END_REF]. Interestingly the E6A, E6K and D16A mutations increased the viscosity of these unacetylated proteins to values comparable to acetylated Cdc8. The predicted distances between mutated residues and charged residues on adjacent carboxy termini strands of the overlapping region are too great (> 12 Å) to form inter-chain electrostatic interactions. Interestingly, while there are 4 positive and 5 negatively charged residues on the Cdc8 amino terminus, there are only 5 negative charges on the carboxyl terminus (Figure 1C). Therefore changes in inter-chain interactions caused by mutating residues D2, E6 or D16 are likely to be due to changes in the overall electrostatic interactions between the regions. However each of these single mutations are likely to affect the stabilization of the amino-terminal helix. Indeed it has been predicted that the amino terminus of muscle Tm is stabilized by interactions between lysine 6 and acetylated methionine 1 [11]. This is consistent with the results presented here for the Cdc8-E6K mutation, which would mimic the stabilizing amino-terminal intra-helical electrostatic interactions observed in vertebrate muscle Tms. It is therefore likely that the observed differences were due to stabilisation of the amino terminal coil structure (through the formation of intra-helical hydrogen bonds) and changes in the overall charge of the amino terminus.

Perhaps unsurprisingly, the LKL mutation had the most dramatically effect upon viscosity illustrating this protein was capable of forming still longer polymers, which is consistent with the molecular models which indicate the A11L mutation is likely to have a significant effect upon the stability of the amino-terminal helical structure. The EM data also indicated that acetylation and the LKL mutation stiffened the Cdc8 protein, as reflected by an increased persistence length when compared to unacetylated Cdc8.

The ability of Cdc8 to form filaments was largely reflected in the ability of Cdc8 to bind to actin. The D2A protein, of which the amino terminal methionine had been cleaved by the E. coli expression system, was unable to form filaments (Figure 3A) and subsequently unable bind actin, consistent with the cooperative binding model [START_REF] Holmes | Gestalt-binding of tropomyosin to actin filaments[END_REF] of Tm binding to actin. The unacetylated E6A and E6K mutants which were able to form filaments as readily as Cdc8 ACE correspondingly had an affinity for actin comparable to the acetylated protein. Interestingly the D16A mutant, also able to form filaments as readily as Cdc8 ACE , did not show a comparable increase in actin affinity. This increase in propensity to form filaments suggests the ability to form filaments can be affected by more than the Tm overlap region, as predicted.

Surprisingly the LKL mutant was found to have a weak affinity for actin. It is unlikely mutation of alanine 13 in the f position of the heptad to an evolutionarily conserved leucine would inhibit interactions between Tm and actin. Tm dimers are thought to have a natural curvature that facilitates their binding to actin [START_REF] Brown | Structure of the mid-region of tropomyosin: bending and binding sites for actin[END_REF][START_REF] Lorenz | An atomic model of the unregulated thin filament obtained by X-ray fiber diffraction on oriented actin-tropomyosin gels[END_REF]. This reduction in actin affinity may therefore be due to the mutation changing the shape of the Tm polymer. Alternatively the propensity of the Cdc8-LKL to form longer filaments (consistent with the viscometry and EM data, together with its ability to sediment in the absence of actin, and the modeling) may inhibit its interaction with the actin polymer. This is consistent with a cooperative model of Tm binding [START_REF] Holmes | Gestalt-binding of tropomyosin to actin filaments[END_REF], in which the overall shape of the Tm dimer is important for actin binding and that inherent helical contour of Tm dimer are an essential feature that facilitates the binding of the elongated protein on actin. Acetylated Cdc8 binding to actin was able to inhibit the actin induced ATPase activity of myosin sub-fragment 1 (S1), and is consistent with reported findings that Cdc8 reduced fission yeast Myo2-driven actin filament gliding in motility assays [START_REF] Stark | Tropomyosin and myosin-II cellular levels promote actomyosin ring assembly in fission yeast[END_REF]. The unacetylated Cdc8, which has weaker endto-end interactions, lower affinity for actin (Figures 3A and4D) and a higher occupancy of the 'open' position on actin [15], had a correspondingly reduced inhibitory effect on S1 ATPase rate (table 2). The inhibition of S1 ATPase rate by Cdc8-E6A, Cdc8-E6K and D16A reflected their affinity for actin. The E6A and E6K proteins, which had an affinity for actin comparable to the acetylated protein (Figure 4D), were found to inhibit ATPase rate correspondingly (Table 2). Where as the Cdc8-D16A protein, which had a lower affinity for actin, correspondingly had less of an inhibitory effect on ATPase activity.

In contrast the Cdc8-LKL protein only weakly inhibited S1 ATPase rate (table 2). Different Tms occupy different positions on actin filaments [27, 28] and changes in one part of the molecule appear to have significant effects further along. One hypothesis is that the substitution of 3 amino acids (ARA to LKL) alters the molecule in such a way that it occupies a more 'open' position on the actin filament, and as a result is less able to inhibit myosin ATPase rate. The small changes to the charge and/or size of residues within the amino terminus dramatically altered the biochemical properties of the proteins in vitro, however, with one exception, each mutant protein (Cdc8-D2A*) was capable of complementing the function of the temperature sensitive cdc8-110 allele within S. pombe. However, it has yet to be assessed whether the amino terminal residue is cleaved from the Cdc8-D2A protein within the yeast cell.

Why is the conserved LKL motif replaced with ARA residues in S. pombe Cdc8? This replacement is conserved, and there are only minor differences in lengths of residue side chains, but this replacement has a surprising dramatic effect on the protein's ability to polymerise in vitro. Molecular modelling predicts the A11L mutation would fill a hydrophobic pocket present in the wild type Cdc8 overlap region, and therefore stabilises the overlap of Tm dimers. Potentially the ARA residues allow acetylation to regulate the protein more than would otherwise be possible if Cdc8 were to possess the LKL motif. Unlike other yeast and metazoan cells, S. pombe expresses a single Tm isoform. Therefore the ability to modulate the physical properties of this sole Tm is likely to be crucial for the growth and viability of the fission yeast

Biochemical

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  F U = " K + K 2 + 8KP 0 4 P 0constant and the average efflux time, calculated from 5 observations per sample at each NaCl concentration.

Biochemical

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  Biochemical Journal Immediate Publication. Published on 09 Jun 2011 as manuscript BJ20101316 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20101316

Biochemical

  Journal Immediate Publication. Published on 09 Jun 2011 as manuscript BJ20101316 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20101316 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2011 The Authors Journal compilation © 2011 Portland Press Limited from at least 3 independent experiments. Averages and standard deviations for WT and Cdc8 mutants were calculated and are summarised in Table

Biochemical

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Biochemical

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Biochemical

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Figure 1 .

 1 Figure 1. (A) Diagram illustrating the heptad repeats within two α-helices highlighting hydrophobic interactions between residues in the a and d positions as well as ionic interactions (dotted lines) between residues in e and g positions. (B) Sequence alignment of 20 amino terminal amino acid residues from fission yeast, budding yeast and mammalian tropomyosins, highlighting position in the heptad repeat (lower case), conserved residues (grey highlight) and positions of mutations introduced in this study (bold). (C) Model of the Cdc8 overlap complex displayed as ribbons (upper panel) or surface (lower panel) with the C termini in green, N termini in orange. The D2, E6, D16 residues and ARA motif are labelled in the ribbons form. The surface representation displays charges within the overlap region; Asp and Glu residues are highlighted in red; Arg and Lys in blue. (D) Surface representations of models of the wild type Cdc8 (upper panel) and LKL mutant (lower panel). Reside 11 (Ala in wild type and Leu in the mutant) is highlighted in pink -note that the mutant Leu side chain fills a hydrophobic pocket present in the wild type. (E) Coomassie stained SDS-PAGE gel showing purified WT and mutant Cdc8 proteins. (F) cdc8-110 cells expressing WT and mutant cdc8 alleles under the control of the nmt41 promoter grown on EMM agar, lacking leucine and containing thiamine at 36 0 C.

Figure 2 .

 2 Figure 2. (A) Circular dichroism spectrum for unacetylated Cdc8. (B) Differential absorbance data for unacetylated Cdc8, Cdc8-D2A, and Cdc8-LKL at 222 nm. (C) First derivative plots for unacetylated Cdc8, Cdc8-D2A, and Cdc8-LKL at 222 nm Figure 3. (A) Viscosity of 20 µM Cdc8 proteins and buffer control at increasing NaCl concentration (0 to 200 mM) at 23 0 C. (B) Electron micrographs of rotary shadowed Cdc8, Cdc8 ACE and Cdc8-LKL proteins. The longest typical particles for each sample are shown. (C) Polymer lengths of Cdc8, Cdc8ACE and Cdc8-LKL calculated from electron micrographs (n>200).

Figure 4 .

 4 Figure 4. SDS-PAGE gels of the pellets and supernatants from cosedimentation experiments of (A) Cdc8-D2A and (B) Cdc-E6K with actin over increasing tropomyosin concentrations in the presence of 100 mM KCl. (C) Cdc8-LKL pellets and supernatants in the absence of actin. (D) Binding curves of free Cdc8 concentration against ratio of density of actin, for WT and mutants Cdc8, measured by densitometry of co-sedimentation SDS-PAGE gels. Curves represent Hill equation lines of best fit.

Figure 5 .

 5 Figure 5. (A) ATPase rate against S1 concentration for WT and mutant Cdc8 proteins. Dashed grey lines correspond to the minimum (S1) and maximum (S1+actin) rate controls, black lines represent rates in the presence of each Cdc8 protein. Actin concentration was 1 µM. Cdc8 dimer concentrations were 1 µM for Cdc8 ACE and Cdc8-E6A and 2.5 µM for Cdc8. (B) Binding curves for Cdc8-LKL and Cdc8 ACE calculated from ATPase measurements with 1 µM actin, 10 µM S1 and increasing concentrations of Cdc8. Curves represent Hill equation lines of best fit.

Figure 1 Biochemical

 1 Figure 1

Figure 2 Biochemical

 2 Figure 2

Figure 3 Biochemical

 3 Figure 3

Figure 4 Biochemical

 4 Figure 4

Figure 5 Biochemical

 5 Figure 5

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Table 1 -

 1 Oligonucleotides used in this study.

	ab Stock Number	Name	Sequence
	197	cdc8D2AJC20F 5´ GAG ATA TAC ATA TGG CCA AGC TTA GAG AG 3´
	198	cdc8D2AJC20R 5´ CTC TCT AAG CTT CAT ATG TAT ATC TC 3´
	199	cdc8E6AF	5´ GGA TAA GCT TAG AGC GAA AAT TAA TGC CG 32
	00	cdc8E6AR	5´ CGG CAT TAA TTT TCG CTC TAA GCT TAT CC 32
	01	cdc8D16AF	5´ CGT GCT GAG ACT GCA GAG GCT GTC GC 32
	02	cdc8D16AR 5´ GCG ACA GCC TCT GCA GTC TCA GCA CG 32
	32	cdc8LKLF	5´ GAG AAA ATT ATT GCC TTA AAA CTC GAG ACT GAT GAG GC 32
	33	cdc8LKLR	5´ GCC TCA TCA GTC TCG AGT TTT AAG GCA TTA ATT TTC TC 32
	58	cdc8E6KF	5´ GGA TAA GCT TAG AAA GAA AAT TAA TGC C 32
	59	cdc8E6KR	5´ GGC AAT AAT TTT CTT TCT AAG CTT ATC C 3Ĺ

Table 2 -

 2 Summary of analysis of amino terminal Cdc8 mutant proteins.Determined by electronspray MS; 2 Average of 3 experiments ; 3 K D imerisation = [Monomer] 2 /[Dimer] using Equation (2); 4 @ 25ºC;5 Measured at [20!M] (centistokes);6 Variance < 0.001; 7 using 10!M actin; 8 !mole Pi sec -1 !M S1 -1 ; 9 background S1 rate subtracted;10 Normalised against rate with unacetylated Cdc8; * Determined from ATPase assays Biochemical Journal Immediate Publication. Published on 09 Jun 2011 as manuscript BJ20101316 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20101316 A c c e p t e d M a n u s c r i p t

	Tm protein	Source	Mass (Da) 1	Tm (ºC) 2	K dimerisation (nM) 2,3	"H 2,4	Kinematic 2,5,6 Viscosity	Persistence Length	K 50% (100mM KCl) 2,7	ATPase 2,8,9 activity	Normalise d ATPase 10	% Inhibition of ATPase	Relative growth in cdc8-110 cells
	Cdc8	E. coli	18,964	34.7	11.5 ± 2.9 144.73	1.07	50nm ± 10 2.76 ± 0.22 0.044 ± 0.002	1.00	54.9%	-
	Cdc8 (acetylated ) S. pombe	19,006	35.4	1.3 ± 0.4 171.8	1.15	90nm ± 20 0.46 ± 0.10 0.017 ± 0.001	0.46	81.3%	1.00
	Cdc8-D2A	E. coli	18,788	32.4	135 ± 29 128.3	1.05	-	> 20	-	-	-	0.00
	Cdc8-E6A	E. coli	18,905	34.4	16.3 ± 4.1 143.47	1.13	-	0.32 ± 0.14 0.014 ± 0.003	0.18	87.4%	1.04
	Cdc8-E6K	E. coli	18,963	35.4	19.5 ± 4.7 130.2	1.15	-	0.45 ± 0.10 0.015 ± 0.003	0.15	87.1%	1.22
	Cdc8-D16A	E. coli	18,919	35.9	1.8 ± 0.5 159.8	1.14	-	2.27 ± 0.31 0.028 ± 0.001	0.29	77%	0.80
	Cdc8-LKL	E. coli	19,020	37.3 0.19 ± 0.06 177.37	1.22	90nm ± 20	5.19 *	0.086 ± 0.008	0.90	6%	1.49
	1

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ACKNOWLEDGEMENTS

We thank Nancy Adamek and Sam Lynn for purified actin, as well as Ian Bruce, Mike Geeves, Kay Barr for support and Mike Geeves for comments on the manuscript.

FUNDING

This work was funded by grants from the EU FP6 (NACBO-CT-2004-500804), the BBSRC (BB/F011784/1), as well a Traveling Fellowship from the Company of Biologists. The EM work and analysis was supported by grants from the National Institutes of Health to W.L. (HL86655 and HL36153); D.S. was supported by an NIH training grant (HL007224) to the Whitaker Cardiovascular Institute at Boston University (J.E. Freedman, P.I.). Electron microscope facilities, supported by NIH grant (RR08426) to Roger Craig, at the University of Massachusetts Medical School were used.


Issam Naghmouchi
email: issam.nagh@gmail.com

DYNAMICAL PROPERTIES OF MONOTONE DENDRITE MAPS

Keywords: Mathematics Subject Classification. 37B20, 37B45, 37E99 dendrite map, ω-limit set, minimal set, periodic point, transitive, Li-Yorke pair, pointwise recurrent

We show that for a monotone dendrite map f : D → D, any ω-limit set is either finite or a minimal Cantor set. We also prove that U R(f

denote the sets of periodic points, uniformly recurrent points, recurrent points and the union of all ω-limit sets respectively. Moreover, we prove that the following properties are equivalent: (i) R(f ) = D, (ii) R(f ) = D and (iii) D \ End(D) ⊂ P (f ).

Introduction

This paper is devoted to monotone dendrite maps. The structure of ωlimit set for monotone dendrite maps is studied here. Acosta and Eslami [START_REF] Acosta | On open maps between dendrites[END_REF] proved that any infinite ω-limit set of a dendrite homeomorphism is a Cantor minimal set. Efremova and Makhrova [START_REF] Efremova | The dynamics of monotone maps of dendrites[END_REF] constructed a homeomorphism of the Gehman dendrite having an infinite ω-limit set which is a minimal Cantor set. In [START_REF] Naghmouchi | Dynamic of monotone graph, dendrite and dendroid maps[END_REF], we proved that for a monotone dendrite map, any infinite ω-limit set does not contain periodic points. This paper can be viewed as a continuation of the paper [START_REF] Naghmouchi | Dynamic of monotone graph, dendrite and dendroid maps[END_REF]. We prove that any infinite ω-limit set of monotone dendrite map is a minimal Cantor set (Corollary 1.3), this generalizes Acosta and Eslami result in [START_REF] Acosta | On open maps between dendrites[END_REF]. For a graph map f , Hawete [START_REF] Hawete | Relatively Pointwise recurrent graph map[END_REF] proved that U R(f ) = R(f ). For a monotone dendrite map f , we prove that U R(f ) = R(f ) = Λ(f ) = P (f ).

Oversteegen and Tymchatyn [START_REF] Oversteegen | Recurrent homeomorphisms on R 2 are periodic[END_REF] showed that recurrent homeomorphisms of the plane are periodic. Kolev and Pérouème [START_REF] Kolev | Recurrent surface homeomorphisms[END_REF] proved that recurrent homeomorphisms of a compact surface with negative Euler characteristic are still periodic. In this direction, we prove that every relatively recurrent monotone dendrite map is a homeomorphism where every cut point is periodic (Theorem 1.6 and Corollary 1.7). Before stating our main results, we recall some basic properties of dendrites and monotone maps.

A continuum is a compact connected metric space. A topological space is arcwise connected if any two of its points can be joined by an arc. We use the terminologies from Nadler [START_REF] Nadler | Continuum Theory: An Introduction[END_REF]. An arc is any space homeomorphic to the compact interval [0, 1]. A dendrite D is a locally connected continuum which contains no simple closed curve. Recall that any two distinct points A continuous map from a dendrite into itself is called a dendrite map. Every dendrite has the fixed point property (see [START_REF] Nadler | Continuum Theory: An Introduction[END_REF]); that is every dendrite map has a fixed point.

Let Z + and N be the sets of non-negative integers and positive integers respectively. Let X be a compact metric space with metric d. For a subset A of X, denote by A the closure of A and by diam(A) the diameter of A. For δ > 0 and x ∈ X, denote by B(x, δ) := {y ∈ X : d(x, y) < δ}. Let f : X -→ X be a continuous map. Denote by f n the n-th iterate of f ; that is, f 0 = Identity and

f n = f • f n-1 if n ≥ 1. For any x ∈ X the subset O f (x) = {f n (x) : n ∈ Z + } is called the f -orbit of x. A point x ∈ X is called periodic of prime period n ∈ N if f n (x) = x and f i (x) = x for 1 ≤ i ≤ n -1.
We define the ω-limit set of a point x to be the set

ω f (x) = {y ∈ X : ∃ n i ∈ N, n i → ∞, lim i→∞ d(f n i (x), y) = 0} = ∩ n∈N {f k (x) : k ≥ n}. A point x ∈ X is said to be recurrent for f if x ∈ ω f (x). The set ω f (x) is a non-empty, closed and strongly invariant set, i.e. f (ω f (x)) = ω f (x). If ω f (x) is finite then it is a periodic orbit. If ω f m (x) is finite for some m ∈ N then ω f (x) is also finite (see [3] for more details). A subset A ⊂ X is called f -invariant if f (A) ⊂ A. It is called a minimal set of f if it is nonempty, closed, f -invariant
and minimal (in the sense of inclusion) for these properties. If X is a minimal set of f , we say that f is a minimal map; in this case, every f -orbit is dense in X. A point x ∈ X is called uniformly recurrent of f if for any neighborhood U of x there exists N ∈ N such that {f n+i (x) : i = 0, 1, ..., N } ∩ U = ∅ for all n ∈ N. Note that x is uniformly recurrent if and only if O f (x) = ω f (x) is a minimal set (see [START_REF] Block | Dynamics in One Dimension[END_REF], Proposition 5, Chapter V). Let F ix(f ), P (f ), U R(f ), R(f ) and Λ(f ) denote the sets of fixed points, periodic points, uniformly recurrent points, recurrent points and the union of all ω-limit sets respectively. Then we have the inclusion relation Fix(f

) ⊂ P(f) ⊂ UR(f) ⊂ R(f) ⊂ Λ(f).
We say that f is pointwise recurrent (resp. relatively recurrent) if R(f ) = X (resp. R(f ) = X). We say that f is transitive if for any two nonempty open sets U and V in X, there exists n ∈ N such that f n (U ) ∩ V = ∅; or equivalently if there is a point x ∈ X for which ω f (x) = X since X here is a compact metric space (see [START_REF] Kolyada | Some aspects of topological transitivity -a survey[END_REF], Theorem 2.2.2).

A pair (x, y)

∈ X×X is called proximal if lim inf n→∞ d(f n (x), f n (y)) = 0, it is called distal if lim inf n→∞ d(f n (x), f n (y)) > 0.
If lim sup n→∞ d(f n (x), f n (y)) = 0, (x, y) is called asymptotic. A pair (x, y) is called a Li-Yorke pair (of f ) if it is proximal but not asymptotic. We say that f is distal if for any x, y ∈ X with x = y, the pair (x, y) is distal.

Definition 1.1. ([8]) Let X, Y be two topological spaces. A map f : X → Y is said to be monotone if for any connected subset C of Y , f -1 (C) is connected.
Notice that f n is monotone for every n ∈ N when f itself is monotone.

Our main results can be stated as follows:

Theorem 1.2. Let f : D → D be a monotone dendrite map. Then for any x ∈ D, we have:

(i) ω f (x) is a minimal set. (ii) ω f (x) ⊂ P (f ).
Corollary 1.3. Let f : D → D be a monotone dendrite map. Then for any x ∈ D, ω f (x) is either a finite set or a minimal Cantor set. In particular, f is not transitive.

Theorem 1.4. Let f : D → D be a monotone dendrite map. Then U R(f ) = R(f ) = Λ(f ) = P (f ). Corollary 1.5. Let f : D → D be a monotone dendrite map. Then the restriction map f |R(f ) is a distal homeomorphism.
Theorem 1.6. Let f : D → D be a monotone dendrite map. Then the following statements are equivalent: (i) f is pointwise recurrent. (ii) f is relatively recurrent. (iii) every cut point is a periodic point.

So from Corollary 1.5, we have the following:

Corollary 1.7. Let f : D → D be a monotone dendrite map. If f is relatively recurrent, then f is a distal homeomorphism.
This paper is organized as follows: In section 2, we give some preliminaries which are useful for the rest of the paper. In section 3, we give some preparatory results concerning monotone dendrite maps and in section 4, we prove the main results of this paper.

Preliminaries

We need the following results: Theorem 2.1. ([2], Theorem 3.10) Let (X, d) be compact metric space and let f : X → X be a continuous map without Li-Yorke pairs. Then f is minimal when it is transitive. Lemma 2.3. If J is a compact interval and f : J → J is a continuous monotone map, then for any x ∈ J, ω f (x) is either a fixed point or a periodic orbit of period 2. In particular, f has no Li-Yorke pair.

Proof. The proof is trivial. Proof. Since f is continuous and monotone, we have

There is δ > 0 such that if v ∈ D with d(v, b) ≤ δ then [v, a] ⊃ [w, a]. Lemma 2.6. Let [a, b] be a non degenerate arc in a dendrite (D, d). Then there is δ > 0 such that [u, v]∩[a, b] = ∅ for any u, v ∈ D satisfying d(a, u) < δ and d(b, v) < δ. Proof. As a = b, there exist y, z ∈ (a, b) such that z ∈ (y, b). By Lemma 2.5, there is δ > 0 such that if u ∈ B(a, δ) and v ∈ B(b, δ) then [u, z] ⊃ [y, z] and [v, y] ⊃ [z, y], so [u, v] = [u, y] ∪ [y, z] ∪ [z, v] ⊃ [y, z]
f ([x, y]) ⊃ [f (x), f (y)] and f -1 ([f (x), f (y)]) ⊃ [x, y] respectively. Hence, [f (x), f (y)] ⊃ f ([x, y]) and therefore, f ([x, y]) = [f (x), f (y)].
Lemma 2.9. Let (D, d) be a dendrite and f : D → D a monotone dendrite map. Suppose that a ∈ F ix(f ) and let x ∈ D. If for some n ∈ Z + and m ∈ N, (a,

f n (x)] ∩ (a, f m+n (x)] = ∅, then (a, x] ∩ (a, f m (x)] = ∅. Proof. Take z ∈ (a, f n (x)] ∩ (a, f m+n (x)]. Since f n ([a, x]) = [a, f n (x)] and f n ([a, f m (x)]) = [a, f m+n (x)] (Lemma 2.8), there exist y 1 ∈ (a, x] and y 2 ∈ (a, f m (x)] such that f n (y 1 ) = f n (y 2 ) = z. By Lemma 2.8, f n ([y 1 , y 2 ]) = {z} so a / ∈ [y 1 , y 2 ] since z = a. Then necessarily, (a, x] ∩ (a, f m (x)] = ∅. Lemma 2.10. Let f : D → D be a monotone dendrite map, a ∈ F ix(f ) and x ∈ D. If [a, x] ⊂ [a, f (x)] then there exists b ∈ F ix(f ) such that lim n→+∞ f n (x) = b and [a, f n (x)] ⊂ [a, b] for all n ∈ Z + .
Proof. We prove by induction on n that [a,

f n (x)] ⊂ [a, f n+1 (x)] for every n ∈ Z + : For n = 0, we have [a, x] ⊂ [a, f (x)]. Suppose that for some n ∈ Z + , [a, f n (x)] ⊂ [a, f n+1 (x)] then by Lemma 2.8, [a, f n+1 (x)] ⊂ [a, f n+2 (x)].
Thus the closure I of the set

I = ∪ n∈Z + [a, f n (x)
] is an f -invariant arc and the sequence (f n (x)) n∈Z + is monotone in this arc, so it converges to a fixed point b ∈ I, and we get [a,

f n (x)] ⊂ [a, b] = I, for all n ∈ Z + .
3. Some results

Lemma 3.1. Let f : D → D be a monotone dendrite map. Let a ∈ F ix(f ) and x ∈ D be such that [a, x] ∩ F ix(f ) = {a} and [a, x] ∩ [a, f (x)] = [a, u 1 ]
where u 1 ∈ (a, x). Then the following statements hold:

(i) if f (u 1 ) ∈ [a, u 1 ) then ω f (x) = {a}. (ii) if f (u 1 ) ∈ (u 1 , f (x)] then there exists b ∈ F ix(f ) such that ω f (x) = {b}. Proof. By Lemma 2.8, f ([a, x]) = [a, f (x)], then as u 1 ∈ (a, f (x)], there is u 0 ∈ (a, x] such that f (u 0 ) = u 1 . Denote for all n ∈ N, u n = f n (u 0 ).
Proof of (i). In this case,

u 0 ∈ (u 1 , x] and as [a, x] ∩ F ix(f ) = {a}, then for every y ∈ [a, u 0 ], ω f (x) = {a}. If for some k ∈ Z + , f k (x) ∈ [a, u 0 ], then it is clear that ω f (x) = {a}. Suppose that for all k ∈ Z + , f k (x) /
∈ [a, u 0 ]. In this case, we will see that for each k ∈ N,

(3.1) [u k , f k (x)] ∩ [a, x] = {u k }.
We proceed by induction on k:

For k = 1, [u 1 , f (x)] ∩ [a,
x] = {u 1 } (see Fig. 1).

Figure 1

Now suppose that for some

k ∈ N, [u k , f k (x)] ∩ [a, x] = {u k }. If [u k+1 , f k+1 (x)] ∩ [a, x] {u k+1 }, then there is y ∈ (u k+1 , f k+1 (x)] ∩ [a, u 1 ]. Since f ([a, u 0 ]) = [a, u 1 ] and f ([u k , f k (x)]) = [u k+1 , f k+1 (x)], there exist w ∈ [a, u 0 ] and v ∈ [u k , f k (x)] such that f (w) = f (v) = y. By Lemma 2.8, f ([w, v]) = {y}. Since [u k , f k (x)] ∩ [a, x] = {u k }, we get u k ∈ [w, v] and so f (u k ) = y = u k+1 , a contradiction. Then [u k+1 , f k+1 (x)] ∩ [a, x] = {u k+1 }.
We will prove now that the sets (u k , f k (x)], k ∈ N, are pairwise disjoint. Suppose that there exist i, j ∈ N and z ∈ D such that z ∈ (u i , f

i (x)] ∩ (u i+j , f i+j (x)]. As f i ([u 0 , x]) = [u i , f i (x)] and f i ([u j , f j (x)]) = [u i+j , f i+j (x)], there exist y 1 ∈ [u 0 , x] and y 2 ∈ [u j , f j (x)] such that f i (y 1 ) = f i (y 2 ) = z, so by Lemma 2.8, f i ([y 1 , y 2 ]) = {z}. By (3.1), [u j , f j (x)] ∩ [a, x] = {u j }, and since y 1 ∈ [u 0 , x] ⊂ [a, x], we have u j ∈ [y 1 , y 2 ], thus u i+j = f i (u j ) = z,
a contradiction. We conclude that the sets (u k , f k (x)], k ∈ N are pairwise disjoint, so by Lemma 2.7, we have lim k→+∞ diam([u k , f k (x)]) = 0 and therefore the pair (x, u 0 ) is asymptotic. Hence,

ω f (x) = ω f (u 0 ) = {a}.
Proof of (ii). By Lemma 2.10, there exists b ∈ F ix(f ) such that lim n→+∞ f n (u 1 ) = b and [a,

u n ] ⊂ [a, b] for all n ∈ N. Clearly, [u 1 , b] ∩ F ix(f ) = {b} (Lemma 2.10
). We distinguish three cases:

• Case 1. b ∈ [u 1 , f (x)]: In this case b = u 2 , indeed, we have u 1 ∈ [x, b], so by Lemma 2.8, u 2 = f (u 1 ) ∈ [f (x), b]. Therefore, u 2 ∈ [f (x), b] ∩ [a, b] = {b}, thus u 2 = b and f ([u 1 , b]) = {b}. Now, if the sets (b, f n (x)], for n ∈ Z + ,
∈ F ix(f ), ω f (x) = {b}. • Case 2. b / ∈ [u 1 , f (x)] and f (x) ∈ [u 1 , b]: In this case, we have, by Lemma 2.10, ω f (x) = ω f (f (x)) = {b}. • Case 3. b / ∈ [u 1 , f (x)] and f (x) / ∈ [u 1 , b]: In this case, [b, x] ∩ [b, f (x)] = [b, v] where v ∈ (u 1 , b]. So f (v) ∈ (v, b] (Lemma 2.

10).

Applying Lemma 3.1, (i) to the map f by considering b instead of a and v instead of u, we get ω f (x) = {b}. The proof is complete.

Lemma 3.2. Let f : D → D be a monotone dendrite map. Let a ∈ F ix(f ) and x ∈ D. If ω f (x) is infinite then for every n ∈ N, [a, x] ∩ [a, f n (x)] = [a, u n ] where u n ∈ F ix(f n ). Proof. One can suppose that [a, x] ∩ F ix(f ) = {a}. Let n ∈ N and [a, x] ∩ [a, f n (x)] = [a, u n ]. Then, u n ∈ [a, x), indeed, if u n = x, then by Lemma 2.10 applied to f n , we get ω f n (x) = {b} ⊂ F ix(f n ) and so ω f (x) is finite, a contradiction. It follows that u n ∈ F ix(f n ) if u n = a and, by Lemma 3.1, if u n ∈ (a, x).

Proof of main results

Proof of Theorem 1.2: If ω f (x) is finite, then Theorem 1.2 obviously holds. In the following we may assume that ω f (x) is infinite.

Proof of (i). Take a ∈ F ix(f ) such that [a, x] ∩ F ix(f ) = {a}. By Lemma 2.7, the sets (a, f 2n (x)], for n ∈ N, cannot be pairwise disjoint (since otherwise, ω f 2 (x) = {a} and so ω f (x) is finite). So by Lemma 2.9, there exist

n 0 ∈ N with n 0 > 1 and u 0 ∈ D such that [a, x] ∩ [a, f n 0 (x)] = [a, u 0 ] where u 0 ∈ (a, x]
. By Lemma 2.10, u 0 ∈ (a, x) and by Lemma 3.2, u 0 ∈ F ix(f n 0 ). If we consider now the map f n 0 then in the same way, we can prove that there exist an integer n 1 ∈ N with n 1 > 1 and a fixed point

u 1 of f n 0 n 1 in the arc (u 0 , x) such that [a, x] ∩ [a, f n 0 n 1 (x)] = [a, u 1 ]
. By induction, we find a sequence of integers (n i ) i∈Z + and a sequence of points (u i ) i∈Z + in D such that for every i ∈ Z + , we have:

(4.1) n i > 1, (4.2)
u i ∈ F ix(f N i ), (4.3)
u i+1 ∈ (u i , x), (4.4) [a, x] ∩ [a, f N i (x)] = [a, u i ],
where N i = Π 0≤j≤i n i . Then by (4.3), the sequence of points (u i ) i∈N is monotone in the arc [a, x] so it converges to a point u ∞ ∈ [a, x] (see Figure 2). It is possible that u ∞ = x.

From (4.4), the sets [u i , f N i (x)], for i ∈ N are pairwise disjoint, hence by Lemma 2.7, lim i→+∞ diam([u i , f N i (x)]) = 0. Then lim i→+∞ f N i (x) = u ∞ and by (4.1), lim i→+∞ N i = +∞ so u ∞ ∈ ω f (x). If u ∞ = x, then the map

f |ω f (x) is transitive. If u ∞ = x, then for every i ∈ N, we have [u ∞ , x] ⊂ [u i , x] and since f N i ([u i , x]) = [u i , f N i (x)] (Lemma 2.8), [f N i (u ∞ ), f N i (x)] = f N i ([u ∞ , x]) ⊂ [u i , f N i (x)]. So lim i→+∞ diam([f N i (u ∞ ), f N i (x)]) = 0.
Then the pair (u ∞ , x) is proximal and by Theorem 2.2, it is an asymptotic pair, so ω f (u ∞ ) = ω f (x). Therefore, regardless of whether u ∞ = x or u ∞ = x, f |ω f (x) is a transitive map without Li-Yorke pairs, hence, by Theorem 2.1, this map is minimal and so ω f (x) is a minimal set.

Proof of (ii): By (4.2), u i ∈ P (f ) for all i ∈ N. Since lim

i→+∞ u i = u ∞ , we have u ∞ ∈ P (f ). As f (P (f )) = P (f ), then ω f (x) = ω f (u ∞ ) ⊂ P (f ).
Proof of Corollary 1.3: Let x ∈ D be such that ω f (x) is infinite. By Theorem 1.2, ω f (x) is minimal so it has no isolated point. To prove that ω f (x) is a Cantor set, it suffices to prove that it is totally disconnected: Otherwise, ω f (x) contains a non-degenerate arc [a, b]. By Theorem 1.2, a, b ∈ P (f ), so by Lemma 2.6, there are p, q ∈ P (f ) such that [p, q] ∩ [a, b] = ∅. Take y ∈ [p, q] ∩ [a, b]. Since p and q are periodic, there is n ∈ N such that f n (p) = p and f n (q) = q, so f n ([p, q]) = [p, q] (Lemma 2.8). By Lemma 2.3, ω f n (y) is finite and so is ω f (y). But this contradicts that ω f (y) = ω f (x) is an infinite minimal set. Thus ω f (x) must be totally disconnected.

Proof of Theorem 1.4: By Theorem 1.2, we have U R(f ) = R(f ) = Λ(f ) ⊂ P (f ). So it suffices to prove that P (f ) ⊂ R(f ). Let x ∈ P (f ). We distinguish two cases:

• Case 1. ω f (x) is a periodic orbit: Without loss of generality, one can assume that ω f (x) = {a} ⊂ F ix(f ). We will prove that x = a and so x ∈ R(f ). Suppose that x = a. Take w ∈ (x, a), by Lemma 2.5, there is δ > 0 such that if p ∈ B(x, δ) then [p, a] ⊃ [w, a]. As x ∈ P (f ), one can choose p ∈ P (f ). We will show that [p, a] ⊃ [x, a]: otherwise, [p, a] ∩ [x, a] = [v, a] where v ∈ (x, a), so take v ∈ (x, v), then by Lemma 2.5 applying for v instead of w, there is

q ∈ P (f ) such that [q, a] ⊃ [v , a], hence [q, a] ∩ [p, a] = [v, a]. Thus, a / ∈ [p, q] (since otherwise, [q, a] ∩ [p, a] = {a}, but [q, a] ∩ [p, a] = [v, a] = {a}) and v ∈ [p, q].
As p and q are periodic points, there is n ∈ N such that f n (p) = p and f n (q) = q, so by Lemma 2.8, f n ([p, q]) = [p, q] and hence ω f n (v) ⊂ [p, q]. As ω f (x) = {a}, the pair (x, a) is asymptotic and then, by Lemma 2.4, lim n→+∞ diam([f n (x), a]) = 0. Hence, for any point y ∈ [x, a], ω f (y) = {a}. In particular, ω f (v) = {a} and then ω

f n (v) = {a}, a contradiction since a / ∈ [q, p]. Thus, [p, a] ⊃ [x, a]. The arc [p, a] is f n -invariant, by Lemma 2.8, so f n (x) ∈ [p, a]. In fact, since ω f (x) = {a}, f n (x) ∈ (x, a] and there is x -n ∈ (p, x) such that f n (x -n ) = x. So for each m ∈ N, ω f m (x -n ) = {a}. Again since x ∈ P (f ) and x = a, one can find q ∈ P (f ) such that [q, p] ⊃ [x -n , p] and a / ∈ [q, p]. Take m ∈ N such that f m (q) = q and f m (p) = p, then, by Lemma 2.8, f m ([q, p]) = [q, p]. So ω f m (x -n ) ⊂ [q, p
], a contradiction. Therefore, x = a.

• Case 2. ω f (x) is infinite: Let u ∞ and u 0 ∈ P (f ) given in the proof of Theorem 1.2. We will prove that x = u ∞ , and so x ∈ ω f (x) which implies that x ∈ R(f ): Assume that x = u ∞ . As x ∈ P (f ), then by Lemma 2.5, we can find p ∈ P

(f ) such that [p, u 0 ] ⊃ [u ∞ , u 0 ], so u ∞ ∈ [p, u 0 ]. Take n ∈ N such that f n (p) = p and f n (u 0 ) = u 0 , hence by Lemma 2.8, f n ([p, u 0 ]) = [p, u 0 ] , so, by Lemma 2.3, ω f n (u ∞ ) is finite and so is ω f (u ∞ ) = ω f (x), a contradiction.
For both cases 1 or 2, we proved that x ∈ R(f

). So P (f ) ⊂ R(f ) and therefore U R(f ) = R(f ) = Λ(f ) = P (f ).
Proof of Corollary 1.5: Take x, y ∈ R(f ) with x = y. By Theorem 2.2, the pair (x, y) is either asymptotic or distal. Let us prove that the pair (x, y) is distal. Indeed, suppose that (x, y) is an asymptotic pair, then by Lemma 2.4, lim n→+∞ diam([f n (x), f n (y)]) = 0 and thus, ω f (x) = ω f (y) = ω f (z) for any z ∈ [x, y]. As x, y ∈ P (f ) (Theorem 1.4), then, by Lemma 2.6, there exist p, q ∈ P (f ) such that [p, q] ∩ [x, y] = ∅. Take z ∈ [p, q] ∩ [x, y]. Let n ∈ N be such that f n (p) = p and f n (q) = q, hence by Lemma 2.8, the arc [p, q] is f n invariant. So by Lemma 2.3, ω f n (z) is finite, hence it is a periodic orbit for f n and so ω f (z) = ω f (x) = ω f (y) is a periodic orbit for f . As x, y ∈ R(f ) we have x, y ∈ P (f ) which is impossible. Therefore, the map f |R(f ) is one to one. Moreover, as f (R(f )) = R(f ) and R(f ) = P (f ) is compact, the map f |R(f ) is a homeomorphism.

  x, y of a dendrite D can be joined by a unique arc with endpoints x and y, denote this arc by [x, y] and let [x, y) = [x, y] \ {y} (resp. (x, y] = [x, y] \ {x} and (x, y) = [x, y] \ {x, y}). A point e ∈ D is called an endpoint if D \ {e} is connected. The set of endpoints of D is denoted by End(D). Any point x ∈ D \ End(D) is called a cut point. The set of cut points of D is dense in D.

Theorem 2 . 2 .

 22 ([12], Corollary 1.2) Let f : D → D be a monotone dendrite map. Then f has no Li-Yorke pairs.

Lemma 2 . 4 .

 24 ([10], Lemma 2.1) Let (D, d) be a dendrite. Then for every ε > 0, there exists δ = δ(ε) > 0 such that, for any x, y ∈ D with d(x, y) ≤ δ, we have diam([x, y]) < ε. Lemma 2.5. ([10], Lemma 2.2) Let [a, b] be an arc in a dendrite (D, d) and w ∈ [a, b).

  and hence [u, v] ∩ [a, b] = ∅. This completes the proof. Lemma 2.7. ([10], Lemma 2.3) Let (C i ) i∈N be a sequence of connected subsets of a dendrite (D, d). If C i ∩ C j = ∅ for all i = j, then lim n→+∞ diam(C n ) = 0. Lemma 2.8. Let (D, d) be a dendrite and f : D → D a monotone dendrite map. Then for any x, y ∈ D, f ([x, y]) = [f (x), f (y)].

  are pairwise disjoint then by Lemma 2.7, lim n→∞ diam([b, f n (x)]) = 0 and so ω f (x) = {b}. Otherwise, there exist n ∈ Z + and m ∈ N such that (b, f n (x)]∩(b, f m+n (x)] = ∅. So by Lemma 2.9,[b, x] ∩ [b, f m (x)] = [b, v] for some v ∈ (b, x]. Let us show that v ∈ [b, u 1 ]: We have b ∈ [a, f (x)],and since b ∈ F ix(f ), we have b ∈ [a, f m (x)] (Lemma 2.8). So {a, b, v} ⊂ [a, f m (x)], then v does not belong to (u 1 , x] (since otherwise the set {a, b, v} cannot be included in an arc). We have v ∈ (b, u 1 ] and f ([u 1 , b]) = {b} then f m (v) = b. In result, [b, x] ∩ [b, f m (x)] = [b, v] where v ∈ (b, u 1 ] ⊂ (b, x) and f m (v) = b. Applying Lemma 3.1,(i) to the map f m by considering the fixed point b of f m instead of a and the point v instead of u 1 , we get ω f m (x) = {b} and as b

Proof of Theorem 1 . 6 :

 16 (i) ⇒ (ii) is clear. (iii) ⇒ (i) follows from Theorem 1.4. It remains to prove (ii) ⇒ (iii): By Theorem 1.4, P (f ) = D. Take x a cut point of D, then D \ {x} has more than one connected component and let A and B be two disjoint connected components of D \ {x}. Since the set P (f ) is dense, one can find two periodic points a and b with a ∈ A and b ∈ B, so x ∈ [a, b]. Without loss of generality, one can assume that a, b ∈F ix(f ). Suppose that x is not a fixed point. If f (x) ∈ (x, b] then f n (x) ∈ [f (x), b] for all n ∈ N. Take w ∈ (x, f (x)). As w = f (x) and P (f ) = D, there is, by Lemma 2.5, a periodic point p such that [f (x), b] ∩ [p, a] = ∅ and [p, a] ⊃ [w, x], so x ∈ [p, a]. Let m ∈ N be the period of p, then f m ([p, a]) = [p, a] (Lemma 2.8). Therefore, f km (x) ∈ [p, a] for all k ∈ Z + and we have f n (x) ∈ [f (x), b] for all n ∈ N, this is a contradiction since [f (x), b] ∩ [p, a] = ∅.By a similar way, the case f (x) ∈ [a, x) leads to a contradiction. Therefore, x ∈ P (f ), this completes the proof.

Acknowledgements. I would like to thanks the referees for their useful comments.

 [START_REF] Acosta | On open maps between dendrites[END_REF]This work was supported by the research unit 99UR/15-15 2000


Issam Naghmouchi
email: issam.nagh@gmail.com

POINTWISE-RECURRENT DENDRITE MAPS

Keywords: Mathematics Subject Classification. 37E99, 37B20 dendrite, dendrite map, ω-limit set, minimal set, periodic point, recurrent point, regularly recurrent point, pointwise-recurrent map

Let D be a dendrite and f : D → D a continuous map.

Denote by E(D) and B(D) the sets of endpoints and branch points of D respectively. We show that if E(D) is countable (resp. B(D) is discrete) then f is pointwise-recurrent if and only if f is pointwise periodic homeomorphism (resp. every point in D\E(D) is periodic).

Introduction

Recurrence and periodicity play an important role in studying dynamical systems. It is interesting to study maps f : X → X from a topological space X to itself that are pointwise-periodic (i.e. all points in X are periodic), or pointwise-recurrent (i.e. all points in X are recurrent). Montgomery [START_REF] Montgomery | Pointwise periodic homeomorphisms[END_REF] showed that if X is a connected topological manifold, a pointwise-periodic homeomorphism f : X → X must be periodic. Weaver [START_REF] Weaver | Pointwise periodic homeomorphisms of continua[END_REF] showed that if X is a continuum embedded in an orientable 2-manifold, an orientationpreserving C 1 -homeomorphism f : X → X has this property. Gottschalk [START_REF] Gottschalk | Powers of homeomorphisms with almost periodic properties[END_REF] proved that if X is a continuum then relatively recurrent homeomorphism f : X → X(i.e. the closure of recurrent point set is dense in X) has every recurrent cut point periodic. In [START_REF] Oversteegen | Recurrent homeomorphisms on R 2 are periodic[END_REF], Oversteegen and Tymchatyn showed that recurrent homeomorphisms of the plane are periodic. Kolev and Pérouème [START_REF] Kolev | Recurrent surface homeomorphisms[END_REF] proved that recurrent homeomorphisms of a compact surface with negative Euler characteristic are still periodic. Recently, Mai [START_REF] Mai | Pointwise-recurrent graph maps[END_REF] showed that a graph map f : G → G is pointwise-recurrent if and only if one of the following statements holds:

(1) G is a circle, and f is a homeomorphism topologically conjugate to an irrational rotation [START_REF] Arevalo | Dendrites with a closed set of end points[END_REF] f is a periodic homemorphism.

In this paper we will study pointwise-recurrent dendrite maps, their dynamical behaviors are both important and interesting in the study of dynamical systems and continuum theory. Recent interest in dynamics on dendrites is motivated by the fact that dendrites have often appear as Julia sets in complex dynamics (see [START_REF] Beardon | Iteration of Rational Functions[END_REF]). In ( [START_REF] Acosta | On open maps between dendrites[END_REF], [START_REF] Balibrea | Minimal sets on graphs and dendrites[END_REF], [START_REF] Efremova | The dynamics of monotone maps of dendrites[END_REF], [START_REF] Mai | R = P for maps of dendrites X with Card(End(X)) < c[END_REF], [START_REF] Naghmouchi | Dynamic of monotone graph, dendrite and dendroid maps[END_REF] and [START_REF] Naghmouchi | Dynamical properties of monotone dendrite maps[END_REF]) several results concerning dendrites were obtained. In [START_REF] Naghmouchi | Dynamical properties of monotone dendrite maps[END_REF], we proved that every relatively recurrent monotone dendrite map have all its cut points periodic.

Before stating our main results, we recall some basic properties of dendrites and dendrite maps.

A continuum is a compact connected metric space. A topological space is arcwise connected if any two of its points can be joined by an arc. We use the terminologies from Nadler [START_REF] Nadler | Continuum Theory: An Introduction[END_REF]. An arc is any space homeomorphic to the compact interval [0, 1]. By a dendrite D, we mean a locally connected continuum which contains no homeomorphic copy to a circle. Every sub-continuum of a dendrite is a dendrite ( [START_REF] Nadler | Continuum Theory: An Introduction[END_REF], Theorem 10.10) and every connected subset of D is arcwise connected ( [START_REF] Nadler | Continuum Theory: An Introduction[END_REF], Proposition 10.9). In addition, any two distinct points x, y of a dendrite D can be joined by a unique arc with endpoints x and y, denote this arc by [x, y] and let denote by Let Z + and N be the sets of non-negative integers and positive integers respectively. Let X be a compact metric space with metric d and f : X -→ X be a continuous map. Denote by f n the n-th iterate of f ; that is, f 0 = id X : the Identity and

[x, y) = [x, y] \ {y} (resp. (x, y] = [x, y] \ {x} and (x, y) = [x, y] \ {x, y}). A point x ∈ D is called an endpoint if D \ {x} is connected. It is called a branch point if
f n = f • f n-1 if n ≥ 1. For any x ∈ X the subset O f (x) = {f n (x) : n ∈ Z + } is called the f -orbit of x. A point x ∈ X is called periodic of prime period n ∈ N if f n (x) = x and f i (x) = x for 1 ≤ i ≤ n -1. A subset A of X is called f -invariant if f (A) ⊂ A. It is called a minimal set of f if it is non-empty, closed, f -invariant
and minimal (in the sense of inclusion) for these properties. For a subset A of X, denote by A the closure of A and by diam(A) the diameter of A. We define the ω-limit set of a point x to be the set

ω f (x) = {y ∈ X : ∃ n i ∈ N, n i → ∞, lim i→∞ d(f n i (x), y) = 0} = ∩ n∈N {f k (x) : k ≥ n}.
The set ω f (x) is a non-empty, closed and strongly invariant set, i.e. f (ω f (x)) = ω f (x). A point x ∈ X is said to be:

-

recurrent for f if x ∈ ω f (x).
almost periodic if for any neighborhood U of x there exists N ∈ N such that {f n+i (x) : i = 0, 1, . . . , N } ∩ U = ∅ for all n ∈ N.

regularly recurrent if for any ε > 0, there is

N ∈ N such that d(x, f kN (x)) < ε for all k ∈ N.
It is easy to see that if x is regularly recurrent then it is almost periodic, hence ω f (x) is a minimal set (see [START_REF] Block | Dynamics in One Dimension[END_REF], Proposition 5, Chapter V).

Let Fix(f ), P(f ), AP(f ) and R(f ) denote the set of fixed points, periodic points, almost periodic and recurrent points respectively. Then we have the following inclusion relation Fix(f

) ⊂ P(f) ⊂ AP(f) ⊂ R(f). We say that f is -pointwise-periodic if P(f ) = X. -pointwise-recurrent if R(f ) = X.
relatively recurrent if R(f) = X.

A continuous map from a dendrite into itself is called a dendrite map.

Our main results are the following:

Theorem 1.1. Let f : D → D be a dendrite map. If B(D) is discrete then f is pointwise-recurrent if and only if f is a homeomorphism and every cut point is periodic.

Following ([2], Corollary 3.6), for any dendrite D, we have B(D) is discrete whenever E(D) is closed. Therefore:

Corollary 1.2. Let f : D → D be a dendrite map. If E(D) is closed then Theorem 1.1 holds. Corollary 1.3. If B(D) is discrete and f : D → D is pointwise-recurrent dendrite map then every endpoint of D is regularly recurrent. Remark 1.4. If E(D) is closed and f : D → D is a pointwise-recurrent dendrite map
, then an endpoint of D may not be in general periodic (see an example by Efremova and Makhrova in [START_REF] Efremova | The dynamics of monotone maps of dendrites[END_REF] on Gehman dendrite).

Theorem 1.5. Let f : D → D be a dendrite map. If E(D) is countable then f is pointwise-recurrent if and only if f is pointwise-periodic homeo- morphism.
Corollary 1.6. [START_REF] Mai | Pointwise-recurrent graph maps[END_REF] Let T be a tree and f : T → T a continuous map. If f is pointwise-recurrent then f is periodic.

Recall that the map f is periodic if f n = id T for some n ∈ N.

Preliminaries

Lemma 2.1. ([14], Lemma 2.3) Let (C i ) i∈N be a sequence of connected subsets of a dendrite (D, d). If C i ∩ C j = ∅ for all i = j, then lim n→+∞ diam(C n ) = 0. Lemma 2.2. Let D be a dendrite and (p n ) n∈N be a sequence of D such that p n+1 ∈ (p n , p n+2 ) for all n ∈ N, and lim n→+∞ p n = p ∞ . Let U n be the connected component of D \ {p n , p ∞ } that contains the open arc (p n , p ∞ ). Then lim n→+∞ diam(U n ) = 0
Proof. It is easy to see that U n+1 ⊂ U n for all n ∈ N. Suppose that Lemma 2.2 is not true, then there is δ > 0 such that for all n ∈ N, diam(U n ) > δ. We will construct an infinite sequence (I n i ) i∈N of pairwise disjoint arcs such that diam(

I n i ) ≥ δ 3 for all i ∈ N which contradicts Lemma 2.1: Take n 0 ∈ N such that for all n ≥ n 0 , diam([p n , p ∞ ]) < δ 3 . For an integer n ≥ n 0 , let a n , b n ∈ U n be such that d(a n , b n ) > δ. There exist c n , d n ∈ (p n , p ∞ ) such that [a n , c n ] ∩ [p n , p ∞ ] = {c n } and [b n , d n ] ∩ [p n , p ∞ ] = {d n }. As d(p n , p ∞ ) < δ 3 , we have either d(c n , a n ) > δ 3 or d(d n , b n ) > δ 3 . So we let I n = [c n , a n ] if d(c n , a n ) > δ 3 and I n = [d n , b n ] if d(d n , b n ) > δ 3 . Choose an integer m > n such that p m ∈ (c n , p ∞ ) if I n = [a n , c n ] and p m ∈ (d n , p ∞ ) if I n = [b n , d n ]. Then I n ∩ U m = ∅: Indeed, otherwise there exists z ∈ I n ∩ U m . Take I n = [a n , c n ]. As c n / ∈ U m and z ∈ U, then [z, c n ] contains p n or p ∞ . In either cases, [z, c n ] ⊃ [c n , p n ] = {c n }, a contradiction.
By the same way as I n , we obtain an arc I m ⊂ U m such that I m intersect the arc [p m , p ∞ ] in a single point and has diameter greater than δ 3 . So by repeating this process infinitely many times beginning from n 0 , we obtain an infinite sequence of arcs (I n i ) i∈N with diameter greater than δ 3 and satisfying the following property: for all i ∈ N, I n i ⊂ U n i and I n i ∩ U n i+1 = ∅. This implies that (I n i ) i∈N are pairwise disjoint, which is our claim.

, m ∈ N such that {f k (x), f m (y)} ⊂ U , then U ∩ P (f ) = ∅.
We say that a dendrite map f : D → D is monotone if the preimage of any point by f is connected. Notice that if f is monotone so is f n for any n ∈ N.

:= f (w -1 ) = z. By continuity of f , we have [z, w] ⊂ f ([a, w -1 ]) ∩ f ([b, w -1 ]) ⊂ f (D).
Hence each point in (z, w) has at least two preimages by f , a contradiction.

(ii) =⇒ (i): Let I ⊂ f (D) be a nondegenerate arc. Then (f -1 ({x})) x∈I is a family of uncountably many pairwise disjoint connected non-empty subsets of D. Suppose that for every x ∈ D, f -1 ({x}) is not reduced to a point, then there is a non degenerate arc I x ⊂ f -1 ({x}) containing no endpoints. By ( [START_REF] Nadler | Continuum Theory: An Introduction[END_REF], Corollary 10.28), D can be written as follow:

D = ∪ n∈N A n ∪ E(D)
where (A n ) n∈N is a family of arcs with pairwise disjoint interiors. Hence, for each arc I x , there is n(x) ∈ N such that I x ∩ A n(x) is a non degenerate arc. So necessarily, there is an arc A n 0 containing an uncountably many pairwise disjoint nondegenerate arcs of (I x ) x∈I , which is a contradiction. Lemma 2.6. Let X be a compact metric space and f : X → X is a pointwise-recurrent continuous map. Then every periodic point of f has a unique pre-image by f n for all n ∈ N.

Proof. As R(f ) = R(f n ) for all n ∈ N (see [START_REF] Block | Dynamics in One Dimension[END_REF]), it suffices to prove the Lemma for f . Suppose that for some periodic point p of period n ∈ N, f -1 ({p}) contains more than one point. So there is q = f n-1 (p) such that f (q) = p. Since q is recurrent and Proof. For n ∈ N, let U n be defined as in Lemma 2.2, let V n := U n ∪{p n , p ∞ } and let denote by N n the period of the point p n . It is easy to see that if there is a sub-sequence of (p n ) n∈N with bounded periods then by the continuity of f (and hence the continuity of its iterated maps), the point p ∞ is periodic, in particular, it is regularly recurrent point. Otherwise, we have to prove the Lemma 2.6 in the case of p ∞ is not periodic. Then without loss of generality, the sequence (p n ) n∈N can be assumed such that

ω f (q) = O f (p), it follows that q ∈ O f (p), so there is k ∈ N such that q = f k (p). Thus f n-1 (p) = f n (q) = f k (f n (p)) = f k (p) = q, a contradiction.
(2.1) (p n , p ∞ ] ∩ Fix(f Nn ) = ∅.
We will prove that for all n ∈ N, the orbit of the point p ∞ under the map f N n+1 is included into the set V n : Indeed, otherwise for some n ∈ N, there is k ∈ N such that f kN n+1 (p ∞ ) / ∈ V n , so we have two possibilities both of them lead to a contradiction:

p ∞ ∈ (p n+1 , f kN n+1 (p ∞ )) or p n ∈ (p n+1 , f kN n+1 (p ∞ )). Let m := kN n+1 . Suppose that p ∞ ∈ (p n+1 , f m (p ∞ )). As p n+1 ∈ Fix(f m ), by the continuity of f m , we have f m ([p n+1 , p ∞ ]) ⊃ [p n+1 , f m (p ∞ )] ∋ p ∞ . Hence there is a point p ∞,-1 ∈ (p n+1 , p ∞ ) such that f m (p ∞,-1 ) = p ∞ . Similarly, there is a point p ∞,-2 ∈ (p n+1 , p ∞,-1 ) such that f m (p ∞,-2 ) = p ∞,-1 . Thus, by induction we construct a sequence (p ∞,-k ) k∈N in (p n+1 , p ∞ ) such that for all k ∈ N, p ∞,-(k+1) ∈ [p n+1 , p ∞,-k ] and (2.2) f km (p ∞,-k ) = p ∞ .
As p ∞,-1 = p ∞ , there is p s ∈ (p ∞,-1 , p ∞ ) for some s ∈ N. Let r ∈ N be such that f r (p n+1 ) = p n+1 and f r (p s ) = p s . Hence, f rm (p n+1 ) = p n+1 and f rm (p s ) = p s . By (2.2), f rm (p ∞,-r ) = p ∞ and by the continuity of f rm ,

f rm ([p n+1 , p ∞,-r ]) ⊃ [p n+1 , p ∞ ] ∋ p s ,
hence p s has a pre-image q by the map f rm in the arc [p n+1 , p ∞,-r ] and as p s / ∈ [p n+1 , p ∞,-r ], q = p s , this contradicts Lemma 2.6 since p s is a fixed point of f rm .

Suppose now the second case,

p n ∈ (p n+1 , f m (p ∞ )). By the continuity of f m , f m ([p n+1 , p ∞ ]) ⊃ [p n+1 , f m (p ∞ )] ∋ p n .
So p n has a preimage q by the map f m in the arc [p n+1 , p ∞ ]. By Lemma 2.6, q is a periodic point that belongs to the orbit of p n hence q has the same period as p n . Hence, q ∈ (p n , p ∞ ] ∩ F ix(f Nn ), this contradict (2.1).

It follows that for any n ∈ N, the orbit of the point p ∞ under the map Assume that f is pointwise-recurrent, then f is surjective. We will use Lemma 2.5: Let I ⊂ D be a nondegenerate arc. Since B(D) is discrete, there exists a non-degenerate open arc J ⊂ I containing no branch points, hence

f N n+1 is included into the set V n and as diam(V n ) = diam(U n ), lim n→+∞ diam(V n ) = 0,
J is an open subset in D. So let x ∈ J. Since f is pointwise-recurrent, one can find n, m ∈ N such that x, f m (x) ∈ J and f n+m (x) ∈ (x, f m (x)). Thus (x, f m (x)) ⊂ I is the connected component of D \ {x, f m (x)} containing the open arc (x, f m (x))
. By Theorem 2.3, (x, f m (x)) contains a periodic point p and by Lemma 2.6, p has a unique preimage by f . We conclude by Lemma 2.5 that f is monotone. Therefore, by ([19], Corollary 1.7), f is a homeomorphism and by Theorem 2.4, every cut point is periodic. The proof is complete. We need the following result.

Theorem 4.1. [14] Let f : D → D be a dendrite map. If E(D) is countable then R(f ) = P (f ). Lemma 4.2. Let f : D → D be a dendrite map. If E(D) is countable and f is pointwise-recurrent then ω f (x) ∩ P (f ) = ∅ for all x ∈ D. Proof. If ω f (x) ⊂ E(D) then ω f (x) is compact countable. If ω f (x) is infinite then it is perfect. Since for any y ∈ ω f (x), y = lim k→+∞ f n k (x) where (n k ) k∈N is an infinite sequence of positif integers. As x ∈ ω f (x), then O f (x) ⊂ ω f (x), hence y is an accumulation point of ω f (x). So ω f (x) is uncountable, a contradiction. Therefore, ω f (x) is finite, that is ω f (x)
is a periodic orbit and so x is a periodic point. One can then assume that ω f (x)\E(D) = ∅, so let y ∈ ω f (x)\E(D). Since R(f ) = D, it follows by Theorem 4.1 that P (f ) = D, so one can find p, q ∈ P (f ) such that y ∈ [p, q]. Let N ∈ N be such that f N (p) = p and f N (q) = q. We already have y ∈ [p, f N (y)] ∪ [q, f n (y)]. One can suppose that y ∈ [p, f N (y)], the same proof being true with p replaced by q. In this case, we have y ∈ f N ([p, y]), so there is y Proof. Let Y be a sub-dendrite of D. By ( [START_REF] Gordh | Monotone Retracts and some Characterizations of Dendrites[END_REF], page 157, (see also [START_REF] Nadler | Continuum Theory: An Introduction[END_REF]) Claim .1. If x ∈ D is not periodic then for any z ∈ D with z = x, (x, z) contains a non periodic branch point. Indeed, [x, z] contains an infinite sequence of branch points that converges to x. Since otherwise, there is y ∈ (x, z] such that (x, y) ∩ B(D) = ∅. So (x, y) is an open subset of D, hence as P (f ) = D, one can find a sequence (p n ) n∈N of periodic points in (x, y) such that p n+1 ∈ (p n , p n+2 ) for all n ∈ N, and lim n→+∞ p n = x. By Lemma 4.3, x is periodic, a contradiction. Now, Suppose that every branch point in (x, z) is periodic, then again by Lemma 4.3, x is periodic. Therefore, necessarily one branch point in (x, z) must be not periodic.

-1 ∈ [p, y] such that f N (y -1 ) = y. Again, y -1 ∈ [p, y] = [p, f N (y -1 )] ⊂ f N ([p, y -1 ]), so there is y -2 ∈ [p, y -1 ] such that f N (y -2 ) = y -1 .
Claim.2. If there exists x ∈ D not periodic then D contains a sub-dendrite with uncountably set of endpoints. Suppose that x ∈ D is not periodic. By Claim.1, there is a ∈ B(D) not periodic. Now, as a is a branch point, there exist two nondegenerate arcs I 0 , I 1 with one endpoint is x and form with [x, a] a family of three arcs having pairwise disjoint interior. As a is not periodic then by Claim.1, there is a 0 ∈ I 0 (resp. a 1 ∈ I 1 ) a non periodic branch point distinct from a. Denote by J 0 (resp. J 1 ) the arc [a, a 0 ] (resp. [a, a 1 ]). In the second step, as a 0 and a 1 are branch points, there exist two nondegenerate arcs I 00 and I 01 (resp. I 10 and I 11 ) with one endpoint is a 0 (resp. a 1 ) and form with [a, a 0 ] (resp. [a, a 1 ]) a family of three arcs having pairwise disjoint interior. Similarly, by Claim.1, we find, for any i, j ∈ {0, 1}, a non periodic branch point a ij ∈ I ij distinct from a i , let denote by J ij := [a i , a ij ]. Thus by induction, we construct for all n ∈ N a sequence of arcs J αn and a sequence of non periodic branch points a αn where α n ∈ {0, 1} n satisfying the following properties: (i) J αn0 = [a αn , a αn0 ] and J αn1 = [a αn , a αn1 ], (ii) J αn ∩ J αn0 = J αn ∩ J αn1 = J αn0 ∩ J αn1 = {a αn }. any n ∈ N, D n = ∪ αn∈{0,1} n J αn , then clearly D n is a tree. Also we have D n+1 ⊂ D n for all n ∈ N. Hence D ∞ = ∪ n∈N D n is a sub-dendrite. Take β ∈ {0, 1} N . For each n ∈ N, we have [a, a β(n) ] ⊂ [a, a β(n+1) ] (where for i ∈ N, we denote by β(i) the first word of length i in the sequence β). Then ∪ n∈N [a, a β(n) ] is an arc where the sequence (a β(n) ) n∈N is monotone in that arc so it converges to a point namely a β that belongs to the dendrite D ∞ . The points a β for β ∈ {0, 1} N are the endpoints of the sub-dendrite D ∞ . Hence, D ∞ has an uncountable set of endpoints .

We deduce then Theorem 1.5 from Claim.2 and Lemma 4.4.

Proof of Corollary 1.6: By Theorem 1.5, f is pointwise-periodic homeomorphism. As E(T ) is finite, there is N ∈ N such that every endpoint of T is fixed by f N . Let a, b ∈ T be two distinct endpoints of T . As f is a homeomorphism (and so is f N ), the arc

  D\{x} has more than two connected components. Denote by E(D) and B(D) the sets of endpoints, and branch points of D respectively. A point x ∈ D \ E(D) is called a cut point. The set of cut points of D is dense in D. A tree is a dendrite with finite set of endpoints.

Theorem 2 . 3 .

 23 ([14], Theorem 2.13) Let f : D → D be a dendrite map, [x, y] be an arc in D, and U be the connected component of D \ {x, y} containing the open arc (x, y). If there exist k

Lemma 2 . 5 .

 25 Theorem 2.4. ([19], Theorem 1.6) Let f : D → D be a dendrite map. If f is monotone then the following statements are equivalent: (i) f is pointwise-recurrent. (ii) f is relatively recurrent. (iii) every cut point is a periodic point. Let f : D → D be a dendrite map, then the following properties are equivalent: (i) any nondegenerate arc I ⊂ f (D) contains a point with unique preimage by f . (ii) the map f is monotone. Proof. (i) =⇒ (ii): If f is not monotone, then there exists z ∈ D such that f -1 (z) is not connected. So one can find a, b ∈ D with a = b and w -1 ∈ (a, b) such that f (a) = f (b) = z and w

Lemma 2 . 7 .

 27 Let f : D → D be a dendrite map. Assume that f is a pointwise-recurrent. Let (p n ) n∈N ⊂ D be a sequence of periodic points of f such that p n+1 ∈ (p n , p n+2 ) for all n ∈ N, and lim n→+∞ p n = p ∞ . Then p ∞ is a regularly recurrent point.

Proof of Corollary 1 . 3 : 4 .

 134 Let e ∈ E(D). By Theorem 1.1, one can find a sequence of periodic points (p n ) n∈N ⊂ D such that p n+1 ∈ (p n , p n+2 ) for all n ∈ N and lim n→+∞ p n = e. Hence by Lemma 2.7, e is regularly recurrent. Proof of Theorem 1.5

  Thus, we construct by induction a sequence (y -n ) n∈N ⊂ [p, y] such that y -(k+1) ∈ (y -k , y -(k+2) ), f N (y -(k+1) ) = y -k for every k ∈ N, and lim n→+∞ y -n = y ∞ ∈ [p, y]. Thereforeone has f N (y ∞ ) = y ∞ (by the continuity of f N ) and so y ∞ ∈ P (f ). Since R(f ) = D and for any k ∈ N, f kN (y -k ) = y, it follows that y -k ∈ ω f (y -k ) = ω f (y). Therefore y ∞ ∈ ω f (y) and as ω f (y) ⊂ ω f (x), we conclude that y ∞ ∈ ω f (x) ∩ P (f ).

Lemma 4 . 3 .

 43 Let f : D → D be a dendrite map and let p ∞ as in Lemma 2.7. If E(D) is countable and f is pointwise-recurrent, then p ∞ is a periodic point. Proof. By Lemma 2.7, p ∞ is regularly recurrent, hence ω f (p ∞ ) is a minimal set. It follows, by Lemma 4.2, that ω f (p ∞ ) is a periodic orbit, and so p ∞ is a periodic point since p ∞ ∈ ω f (p ∞ ).

Lemma 4 . 4 .

 44 Let D be a dendrite with countable set of endpoints. Then every sub-dendrite of D has countable set of endpoints.

  ), Y is a monotone retraction of D by r : D → Y . Then we have r(E(D)) = E(Y ). Indeed, r(E(D)) ⊂ E(Y ) follows from the fact that r(x) lies in any arc joining x to any point of Y . Let a ∈ E(Y ) and let Z the connected component of D \ Y such that Z contains a. Then r(Z) = {a}. As Z ∩ E(D) = ∅, there is b ∈ Z ∩ E(D) with r(b) = a. It follows that Card(E(Y )) ≤ Card(E(D)) and hence E(Y ) is countable. Proof of Theorem 1.5: The "if" part of the Theorem is clear. Lets prove the "only if" part: By Theorem 4.1, one has P (f ) = D.

  [a, b] is f N -invariant, hence every point in [a, b] is fixed by f N . Since T = ∪ a,b∈E(T ) [a, b],every point in the tree T is fixed by f N . So f is periodic.

Acknowledgements. I would like to thanks Professor Habib Marzougui for helpful discussions on the subject of the paper.

This work was supported by the research unit 99UR/ 15-15 2000