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 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.

Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity

1991 ◽  
Vol 11 (1) ◽  
pp. 115-128 ◽  
Author(s):  
J. Llibre ◽  
R. S. Mackay
Keyword(s):  
Periodic Orbit ◽  
Convex Hull ◽  
Topological Entropy ◽  
Rational Point ◽  
Rotation Vector ◽  
Connected Subset ◽  
Positive Topological Entropy ◽  
Rotation Vectors ◽  
Compact Connected Subset ◽  
Rotation Set

AbstractWe show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ∈ ℤ2, q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.

Positive topological entropy for monotone recurrence relations

2014 ◽  
Vol 35 (6) ◽  
pp. 1880-1901 ◽  
Author(s):  
LI GUO ◽  
XUE-QING MIAO ◽  
YA-NAN WANG ◽  
WEN-XIN QIN
Keyword(s):  
Recurrence Relation ◽  
Topological Entropy ◽  
Rotation Number ◽  
Recurrence Relations ◽  
High Dimensional ◽  
Positive Topological Entropy ◽  
Mather Theory

We associate the topological entropy of monotone recurrence relations with the Aubry–Mather theory. If there exists an interval$[{\it\rho}_{0},{\it\rho}_{1}]$such that, for each${\it\omega}\in ({\it\rho}_{0},{\it\rho}_{1})$, all Birkhoff minimizers with rotation number${\it\omega}$do not form a foliation, then the diffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy.

scholarly journals Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

2021 ◽  
pp. 1-40
Author(s):  
ALENA ERCHENKO
Keyword(s):  
Lyapunov Exponent ◽  
Probability Measure ◽  
Lebesgue Measure ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Maximal Entropy ◽  
Anosov Diffeomorphism ◽  
Dynamical Invariants ◽  
Area Preserving ◽  
Measure Of Maximal Entropy

Abstract We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

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 ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

scholarly journals Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1602
Author(s):  
Jan Andres ◽  
Jerzy Jezierski
Keyword(s):  
Differential Equations ◽  
Topological Entropy ◽  
Lower Estimate ◽  
Deterministic Chaos ◽  
Impulsive Differential Equations ◽  
Nielsen Numbers ◽  
Positive Topological Entropy ◽  
Admissible Pairs ◽  
Impulsive Dynamics ◽  
Differential Equations And Inclusions

The main aim of this article is two-fold: (i) to generalize into a multivalued setting the classical Ivanov theorem about the lower estimate of a topological entropy in terms of the asymptotic Nielsen numbers, and (ii) to apply the related inequality for admissible pairs to impulsive differential equations and inclusions on tori. In case of a positive topological entropy, the obtained result can be regarded as a nontrivial contribution to deterministic chaos for multivalued impulsive dynamics.

scholarly journals A Global Analysis of The Generalized Sitnikov Problem

1999 ◽  
Vol 172 ◽  
pp. 291-302
Author(s):  
Steven R. Chesley
Keyword(s):  
Angular Momentum ◽  
Global Analysis ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Bounded Motion ◽  
Type Symmetry ◽  
The Stability ◽  
Two Parameters ◽  
Three Body ◽  
Area Preserving

AbstractThe isosceles three-body problem with Sitnikov-type symmetry has been reduced to a two-dimensional area-preserving Poincaré map depending on two parameters: the mass ratio, and the total angular momentum. The entire parameter space is explored, contrasting new results with ones obtained previously in the planar (zero angular momentum) case. The region of allowable motion is divided into subregions according to a symbolic dynamics representation. This enables a geometric description of the system based on the intersection of the images of the subregions with the preimages. The paper also describes the regions of allowable motion and bounded motion, and discusses the stability of the dominant periodic orbit.

OBSERVED ROTATION NUMBERS IN FAMILIES OF CIRCLE MAPS

2001 ◽  
Vol 11 (01) ◽  
pp. 73-89 ◽  
Author(s):  
MICHAEL A. SAUM ◽  
TODD R. YOUNG
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Narrow Band ◽  
Initial Conditions ◽  
Positive Probability ◽  
Circle Maps ◽  
Numerical Evidence ◽  
Rotation Interval ◽  
Large Numbers ◽  
Rotation Numbers

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers.

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