A condition that implies full homotopical complexity of orbits for surface homeomorphisms

2019 ◽  
Vol 41 (1) ◽  
pp. 1-47
Author(s):  
SALVADOR ADDAS-ZANATA ◽  
BRUNO DE PAULA JACOIA
Keyword(s):  
Compact Convex ◽  
Universal Cover ◽  
Invariant Sets ◽  
Maximal Dimension ◽  
Rotation Vectors ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Poincaré Disk ◽  
Area Preserving ◽  
Uniformly Bounded

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ isotopic to the identity. A set of hypotheses is presented, called a fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}$), denoted by $\widetilde{f},$ has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, for $C^{1+\unicode[STIX]{x1D716}}$ diffeomorphisms, we show the existence of rotational horseshoes having non-trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets and by periodic orbits in the rational case. Also, $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where $f$ is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

scholarly journals Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior

2013 ◽  
Vol 35 (1) ◽  
pp. 1-33 ◽  
Author(s):  
SALVADOR ADDAS-ZANATA
Keyword(s):  
Periodic Point ◽  
Interior Point ◽  
Connected Component ◽  
Empty Interior ◽  
Dehn Twists ◽  
Topologically Mixing ◽  
Rotation Set ◽  
Area Preserving ◽  
Uniformly Bounded

AbstractIn this paper we consider${C}^{1+ \epsilon } $area-preserving diffeomorphisms of the torus $f$, either homotopic to the identity or to Dehn twists. We suppose that$f$has a lift$\widetilde {f} $to the plane such that its rotation set has interior and prove, among other things, that if zero is an interior point of the rotation set, then there exists a hyperbolic$\widetilde {f} $-periodic point$\widetilde {Q} \in { \mathbb{R} }^{2} $such that${W}^{u} (\widetilde {Q} )$intersects${W}^{s} (\widetilde {Q} + (a, b))$for all integers$(a, b)$, which implies that$ \overline{{W}^{u} (\widetilde {Q} )} $is invariant under integer translations. Moreover,$ \overline{{W}^{u} (\widetilde {Q} )} = \overline{{W}^{s} (\widetilde {Q} )} $and$\widetilde {f} $restricted to$ \overline{{W}^{u} (\widetilde {Q} )} $is invariant and topologically mixing. Each connected component of the complement of$ \overline{{W}^{u} (\widetilde {Q} )} $is a disk with diameter uniformly bounded from above. If$f$is transitive, then$ \overline{{W}^{u} (\widetilde {Q} )} = { \mathbb{R} }^{2} $and$\widetilde {f} $is topologically mixing in the whole plane.

Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist

1997 ◽  
Vol 17 (3) ◽  
pp. 575-591 ◽  
Author(s):  
H. ERIK DOEFF
Keyword(s):  
Fixed Point ◽  
Rotation Number ◽  
Rational Number ◽  
Dehn Twist ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Rotation Vectors ◽  
Rotation Set ◽  
Area Preserving

We extend the theory of rotation vectors to homeomorphisms of the two-dimensional torus that are homotopic to a Dehn twist. We define a one-dimensional rotation number and recreate the theory of the homotopic case to the identity case. We prove that if such a map is area preserving and has mean rotation number zero, then it must have a fixed point. We prove that the rotation set is a compact interval, and that if the rotation interval contains two distinct numbers, then for any rational number in the rotation set there exists a periodic point with that rotation number. Finally, we prove that any interval with rational endpoints can be realized as the rotation set of a map homotopic to a Dehn twist.

scholarly journals Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2293-2333
Author(s):  
JINGZHI YAN
Keyword(s):  
Fixed Point ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Dirac Measure ◽  
Lefschetz Index ◽  
Weak Star ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Area Preserving ◽  
Measure Sense

Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_{0}$ with Lefschetz index one. Le Roux conjectured that $z_{0}$ is accumulated by periodic orbits. In this paper, we will approach Le Roux’s conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_{0}$ and if the area of $M$ is finite, then $z_{0}$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_{0}$ is the limit in the weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.

scholarly journals Dynamics of homeomorphisms of the torus homotopic to Dehn twists

2012 ◽  
Vol 34 (2) ◽  
pp. 409-422 ◽  
Author(s):  
SALVADOR ADDAS-ZANATA ◽  
FÁBIO A. TAL ◽  
BRÁULIO A. GARCIA
Keyword(s):  
Lebesgue Measure ◽  
Topological Entropy ◽  
Rotation Number ◽  
Vertical Velocity ◽  
Positive Topological Entropy ◽  
Dehn Twists ◽  
Bounded Motion ◽  
Rotation Set ◽  
Area Preserving ◽  
Uniformly Bounded

AbstractIn this paper, we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential ‘horizontal’ set $K$, invariant under $f$. In other words, if we consider the lift $\hat {f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat {f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland’s conjecture to this setting: if $f$ is area preserving and has a lift $\hat {f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat {f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

scholarly journals Fully essential dynamics for area-preserving surface homeomorphisms

2017 ◽  
Vol 38 (5) ◽  
pp. 1791-1836 ◽  
Author(s):  
ANDRES KOROPECKI ◽  
FABIO ARMANDO TAL
Keyword(s):  
Initial Conditions ◽  
Essential Dynamics ◽  
Empty Interior ◽  
Fixed Point Set ◽  
Higher Genus ◽  
Point Set ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Bounded Sets ◽  
Area Preserving

We study the interplay between the dynamics of area-preserving surface homeomorphisms homotopic to the identity and the topology of the surface. We define fully essential dynamics and generalize the results previously obtained on strictly toral dynamics to surfaces of higher genus. Non-fully essential dynamics are, in a way, reducible to surfaces of lower genus, while in the fully essential case the dynamics is decomposed into a disjoint union of periodic bounded disks and a complementary invariant externally transitive continuum $C$. When the Misiurewicz–Ziemian rotation set has non-empty interior the dynamics is fully essential, and the set $C$ is (externally) sensitive on initial conditions and realizes all of the rotational dynamics. As a fundamental tool we introduce the notion of homotopically bounded sets and we prove a general boundedness result for invariant open sets when the fixed point set is inessential.

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

scholarly journals Ground states and zero-temperature measures at the boundary of rotation sets

2017 ◽  
Vol 39 (1) ◽  
pp. 201-224
Author(s):  
TAMARA KUCHERENKO ◽  
CHRISTIAN WOLF
Keyword(s):  
Ground State ◽  
Line Segment ◽  
Invariant Measures ◽  
Ground States ◽  
Zero Temperature ◽  
Direction Vector ◽  
Compact Metric Space ◽  
Rotation Vectors ◽  
Continuous Potential ◽  
Rotation Set

We consider a continuous dynamical system $f:X\rightarrow X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{m}):X\rightarrow \mathbb{R}^{m}$. We study the set of ground states $GS(\unicode[STIX]{x1D6FC})$ of the potential $\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D6F7}$ as a function of the direction vector $\unicode[STIX]{x1D6FC}\in S^{m-1}$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\unicode[STIX]{x1D6F7}$. In particular, for each $\unicode[STIX]{x1D6FC}$ the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ forms a non-empty, compact and connected subset of a face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ of the rotation set associated with $\unicode[STIX]{x1D6FC}$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any $m\in \mathbb{N}$ examples with an exposed boundary point (that is, $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ is a non-trivial line segment.

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

ROTATION VECTORS FOR TORAL MAPS AND FLOWS: A TUTORIAL

1995 ◽  
Vol 05 (02) ◽  
pp. 321-348 ◽  
Author(s):  
JAMES A. WALSH
Keyword(s):  
Dynamical Systems ◽  
Rotation Number ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Mode Locking ◽  
Rotation Vector ◽  
Higher Genus ◽  
Rotation Vectors ◽  
Dissipative Dynamical Systems ◽  
Rotation Set

This paper is an introduction to the concept of rotation vector defined for maps and flows on the m-torus. The rotation vector plays an important role in understanding mode locking and chaos in dissipative dynamical systems, and in understanding the transition from quasiperiodic motion on attracting invariant tori in phase space to chaotic behavior on strange attractors. Throughout this article the connection between the rotation vector and the dynamics of the map or flow is emphasized. We begin with a brief introduction to the dimension one setting, in which case the rotation vector reduces to the well known rotation number of H. Poincaré. A survey of the main results concerning the rotation number and bifurcations of circle maps is presented. The various definitions of rotation vector in the higher dimensional setting are then introduced with emphasis again placed on how certain properties of the rotation set relate to the dynamics of the map or flow. The dramatic differences between results in dimension two and results in higher dimensions are also presented. The tutorial concludes with a brief introduction to extensions of the concept of rotation vector to the setting of dynamical systems defined on surfaces of higher genus.

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