Let $f$ be a homeomorphism of the closed annulus $A$ that preserves orientation, boundary components and that has a lift $\tilde f$ to the infinite strip $\tilde A$ which is transitive. We show that, if the rotation number of both boundary components of $A$ is strictly positive, then there exists a closed nonempty connected set $Γ\subset\tilde A$ such that $Γ\subset]-\infty,0]\times[0,1]$, $Γ$ is unlimited, the projection of $Γ$ to $A$ is dense, $Γ-(1,0)\subsetΓ$ and $\tilde{f}(Γ)\subset Γ.$ Also, if $p_1$ is the projection in the first coordinate in $\tilde A$, then there exists $d>0$ such that, for any $\tilde z\inΓ,$ $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$
In particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior.
We prove a generalization of the Poincaré-Birkhoff theorem for the open annulus showing that if a homeomorphism satisfies a certain twist condition and the nonwandering set is connected, then there is a fixed point. Our main focus is the study of bounded homeomorphisms of the open annulus. We prove a fixed point theorem for bounded homeomorphisms and study the special case of those homeomorphisms possessing at most one fixed point. Lastly we use the existence of rational rotation numbers to prove the existence of periodic orbits.
Let H be a homeomorphism of the open annulus isotopic to the identity which admits a lift h to the plane without fixed point. We show that h admits a Brouwer line which is a lift of a properly imbedded line joining one end to the other in the annulus or H admits a free essential simple closed curve.
For any irrational number $α$, there exists an ergodic area preserving homeomorphism of the closed annulus which is isotopic to the identitity, admits no compact invariant set contained in the interior of the annulus, and has the rotation number $α$.
We consider which spaces can be realized as the omega limit set of the discrete time dynamical system. This is equivalent to asking which spaces admit a chain transitive homeomorphism and which do not. This leads us to ask for spaces where all homeomorphisms are chain transitive.
In this article we consider homeomorphisms of the open annulus $\mathbb{A}=\mathbb{R}/\mathbb{Z}\times \mathbb{R}$ which are isotopic to the identity and preserve a Borel probability measure of full support, focusing on the existence of non-contractible periodic orbits. Assume $f$ such homeomorphism such that the connected components of the set of fixed points of $f$ are all compact. Further assume that there exists $\check{f}$ a lift of $f$ to the universal covering of $\mathbb{A}$ such that the set of fixed points of $\check{f}$ is non-empty and that this set projects into an open topologica…
This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $μ$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measu…
In this article, we give a growth rate about the number of periodic orbits in the Franks type theorem obtained by the authors \cite{LWY}. As applications, we prove the following two results: there exist infinitely many distinct non-contractible closed geodesics on $\mathbb{R}P^2$ endowed with a Riemannian metric such that its Gaussian curvature is positive, moreover, the number of non-contractible closed geodesics of length $\leq l$ grows at least like $l^2$; and there exist either two or infinitely many distinct non-contractible closed geodesics on Finsler $\mathbb{R}P^2$ with reversibility $…
Let $X$ be a continuum and let $φ:X\rightarrow X$ be a homeomorphism. To construct a dynamical system $(X,φ)$ with interesting dynamical properties, the continuum $X$ often needs to have some complicated topological structure. In this paper, we are interested in one such dynamical property: transitivity. By now, various examples of continua $X$ have been constructed in such a way that the dynamical system $(X,φ)$ is transitive. Mostly, they are examples of continua that are not path-connected, such as the pseudo-arc or the pseudo-circle, or they are examples of locally connected continua (and …
Consider a homeomorphism h of the closed annulus S^1*[0,1], isotopic to the identity, such that the rotation set of h is reduced to a single irrational number alpha (we say that h is an irrational pseudo-rotation). For every positive integer n, we prove that there exists a simple arc gamma joining one of the boundary component of the annulus to the other one, such that gamma is disjoint from its n first iterates under h. As a corollary, we obtain that the rigid rotation of angle alpha can be approximated by homeomorphisms conjugate to h. The first result stated above is an analog of a theorem …
We prove that a homeomorphism of the torus homotopic to the identity whose rotation set is reduced to a single totally irrational vector is chain-recurrent. In fact, we show that pseudo-orbits can be chosen with a small number of jumps, in particular, that the nonwandering set is weakly transitive. We give an example showing that the nonwandering set of such a homeomorphism may not be transitive.
We show that, if $f$ is a homeomorphism of the 2--torus isotopic to the identity, and its lift $\widetilde f$ is transitive, or even if it is transitive outside of the lift of the elliptic islands, then $(0,0)$ is in the interior of the rotation set of $\widetilde f.$ This proves a particular case of Boyland's conjecture.
Let $h$ be a without fixed point lift to the plane of a homeomorphism of the open annulus isotopic to the identity and without wandering point. We show that $h$ admits a $h$-invariant dense open set $O$ on which it is conjugate to a translation and we study the action of $h$ on the compactly connected components of the closed and without interior set ${\bf R}^2 \setminus O$.
In this paper, we study pseudo-rotations of the open annulus, \emph{i.e.} conservative homeomorphisms of the open annulus whose rotation set is reduced to a single irrational number (the angle of the pseudo-rotation). We prove in particular that, for every pseudo-rotation $h$ of angle $ρ$, the rigid rotation of angle $ρ$ is in the closure of the conjugacy class of $h$. We also prove that pseudo-rotations are not persistent in $C^r$ topology for any $r\geq 0$.
In this work we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy and we obtain several applications. For homeomorphisms of the open annulus $\mathbb{A}$ with zero topological entropy, we show that rotatio…
We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincaré-like classification for this class of maps of Jaeger-Stark, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that, in some of the examples we construct, the unique minimal set is …
Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$, as well as the prime end rotation number $\checkρ_\pm$ of $U_\pm$. The purpose of this paper is to show that $\checkρ_\pm$ belongs to $\tildeρ(X)$ for any separating invariant continuum $X$.
We study the relationship between transitivity and topological chaos for homeomorphisms of the two torus. We show that if a transitive homeomorphism of $\mathbb{T}^2$ is homotopic to the identity and has both a fixed point and a periodic point which is not fixed, then it has a topological horseshoe. We also show that if a transitive homeomorphims of $\mathbb{T}^2$ is homotopic to a Dehn twist, then either it is aperiodic or it has a topological horseshoe.
Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$. For any rational number $p/q\in \tildeρ(X)$, $f$ is shown to admit a $p/q$ periodic point in $X$, provided that (1) $X$ consists of nonwandering points or (2) $X$ is an attractor and the frontiers of $U_-$ and $U_+$ coinci…
A homeomorphism f of a manifold M is called H_1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover \tM. Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H_1(M). For a rel pseudo-Anosov map ϕof a compact surface M we show that the following are equivalent: (a) ϕis H_1-transitive, (b) the action of ϕon H_1(M) has spectral radius one, and (c) the lifts of the invariant foliations of ϕto \tM have dense leaves. The proof relies on a characterization of transitivity for twisted \Z^d-extensions of a transitive …
Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in its interior.
Let $\{f_t\}_{t\in(1,2]}$ be the family of core tent maps of slopes $t$. The parameterized Barge-Martin construction yields a family of disk homeomorphisms $Φ_t\colon D^2\to D^2$, having transitive global attractors $Λ_t$ on which $Φ_t$ is topologically conjugate to the natural extension of $f_t$. The unique family of absolutely continuous invariant measures for $f_t$ induces a family of ergodic $Φ_t$-invariant measures $ν_t$, supported on the attractors $Λ_t$. We show that this family $ν_t$ varies weakly continuously, and that the measures $ν_t$ are physical with respect to a weakly continuou…
