For negative-torsion maps on the annulus we show that on every $\mathcal{C}^1$ essential curve there is at least one point of zero torsion. As an outcome, we deduce that the Hausdorff dimension of the set of points of zero torsion is greater or equal 1. As a byproduct, we obtain a Birkhoff's-theorem-like result for $\mathcal{C}^1$ essential curves in the framework of negative-torsion maps.
For a twist map $f$ of the annulus preserving the Lebesgue measure, we give sufficient conditions to assure the existence of a set of positive measure of points with non-zero asymptotic torsion. In particular, we deduce that every bounded instability region for $f$ contains a set of positive measure of points with non-zero asymptotic torsion. Moreover, for an exact symplectic twist map $f$, we provide a simple, geometric proof of a result by Cheng and Sun (see [CS96]) which characterizes $\mathcal{C}^0$-integrability of $f$ by the absence of conjugate points.
We introduce the notion of abstract angle at a couple of points defined by two radial foliations of the closed annulus. We use this notion to give unified proofs of some classical results on area preserving positive twist maps of the annulus by using the Lifting Theorem and the Intermediate Value Theorem.
It is often the case that a covering map of the open annulus is semiconjugate to a map of the circle of the same degree. We investigate this possibility and its consequences on the dynamics. In particular, we address the problem of the classification up to conjugacy. However, there are examples which are not semiconjugate to a map of the circle, and this opens new questions.
We consider holomorphic maps defined in an annulus around $\mathbb R/\mathbb Z$ in $\mathbb C/\mathbb Z$. E. Risler proved that in a generic analytic family of such maps $f_ζ$ that contains a Brjuno rotation $f_0(z)=z+α$, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near $f_0$. In this paper, we obtain the Risler's result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of $\mathbb R/\mathbb Z$. We prove that this operator is hyperbolic, with one unstable direction co…
Consider a continuous surjective self map of the open annulus with degree d > 1. It is proved that the number of Nielsen classes of periodic points is maximum possible whenever f has a completely invariant essential continuum. The same result is obtained in negative degree |d| > 1 and for just forward invariant essential continua, provided that the continuum is locally connected. We also deal with the problem of wether there is a representative of each Nielsen class in the filled set of the invariant continuum. Moreover, if the map extends continuously to the boundary of the annulus and both b…
These are notes for a minicourse given at Regional Norte UdelaR in Salto, Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian Dynamics. We will present Birkhoff and Aubry-Mather theory for the conservative twist maps of the 2-dimensional annulus and focus on what happens close to the Aubry-Mather sets: definition of the Green bundles, link between hyperbolicity and shape of the Aubry-Mather sets, behaviour close to the boundaries of the instability zones. We will also give some open questions. This course is the second part of a minicourse that was begun by R. Potrie. S…
We construct a $C^2$ symplectic twist map f of the annulus that has an essential invariant curve C such that: - C is not differentiable; - the dynamics of f restricted to C is conjugated to the one of a Denjoy counter-example; - C is at the boundary of an instability zone for f.
Translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems. In this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus. The applications include billiards and geodesic flows. Going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps, G.D. Birkhoff was led to the concept of ``regions of instability'' for twist maps. We discuss the closely related notion of instability sets for a…
The twist interval of a twist map on the annulus $A=\mathbb{T}\times [0,1]$ has nonempty interior if $f$ preserves the area, but could be degenerate for general twist maps. In this note, we show that if a twist map $f$ is non-wandering, then the twist interval of $f$ is non-degenerate. Moreover, if there are two disjoint invariant curves of $f$, then their rotation numbers must be different (no matter if they are rational or irrational).
Poincare's last geometric theorem (Poincare-Birkhoff Theorem) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under $f$ at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincare's geometric theorem, our result also has some applications to reversible systems.
We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound $\mathcal O(N^δ)$ where $δ$ is the dimension of the trapped set of the baker's map and $(2 πN)^{-1}$ is the semiclassical parameter, which improves upon the previous result of $\mathcal O(N^{δ+ ε})$. Furthermore, we derive a Weyl upper bound with explicit dependence on the inner radius of the annulus for quantum open baker's maps with Gevrey cutoffs.
In reversible dynamical systems, it is frequently of importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks' theorem on a dichotomy of the number of periodic points of area preserving maps on the annulus to symmetric periodic points of area preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area preserving rever…
We consider the classical problem of area-preserving maps on annulus $\mathbb{A} = S^1 \times [0, 1]$ . Let $\mathcal{M}_f$ be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism $f$ on $\mathbb{A}$. Given any $μ_1$ and $μ_2$ in $\mathcal{M}_f$, Franks \cite{Franks1988}\cite{Franks1992}, generalizing Poincaré's last geometric theorem (Birkhoff \cite{Birkhoff1913}), showed that if their rotation numbers are different, then $f$ has infinitely many periodic orbits. In this paper, we show that if $μ_1$ and $μ_2$ have different actions, even if…
For a $\mathcal{C}^1$ diffeomorphism $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ isotopic to the identity, we prove that for any value $l\in\mathbb{R}$ of the linking number at finite time of the orbits of two points there exists at least a point whose torsion at the same finite time equals $l\in\mathbb{R}$. As an outcome, we give a much simpler proof of a theorem by Matsumoto and Nakayama concerning torsion of measure on $\mathbb{T}^2$. In addition, in the framework of twist maps, we generalize a known result concerning the linking number of periodic points: indeed, we estimate such value for any…
For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency $ω$ of an integrable system by a trigonometric polynomial of degree $N$ perturbation $R_N$ with $\|R_N\|_{C^r}<ε$. We obtain a relation among $N$, $r$, $ε$ and the arithmetic property of $ω$, for which the area-preserving map admit no invariant circles with $ω$.
Let $f:S^1\times [0,1]\to S^1\times [0,1]$ be a real-analytic annulus diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift $\tilde {f}:\mathbb{R}\times [0,1]\rightarrow \mathbb{R}\times [0,1]$ we have ${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$ and that $\tilde{f}$ positively translates points in $\mathbb{R}\times \{1\}$. Let $\tilde{f}_ε$ be the perturbation of $\tilde{f}$ by the rigid horizontal translation $(x,y)\mapsto (x+ε,y)$. We show that for all $ε>0$ sufficiently small we have ${\rm Fix} (\tilde{f}_ε)=\emptyset $. The proof foll…
The key result of this article is key lemma: if a Jordan curve $γ$ is invariant by a given C 1+$α$ -diffeomorphism f of a surface and if $γ$ carries an ergodic hyperbolic probability $μ$, then $μ$ is supported on a periodic orbit. From this Lemma we deduce three new results for the C 1+$α$ symplectic twist maps f of the annulus: 1. if $γ$ is a loop at the boundary of an instability zone such that f |$γ$ has an irrational rotation number, then the convergence of any orbit to $γ$ is slower than exponential; 2. if $μ$ is an invariant probability that is supported in an invariant curve $γ$ with an…
We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the…
We study dynamical properties of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups. These maps are defined on the unit circle (the boundary of the Poincaré disk) by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity point and show that for a family of such maps the cycle property holds: the orbits coincide after finitely many steps. We also show that for an open set of discontinuity points the associated two-dimensional natural extension maps possess global attractors with finite …
We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of $\mathbb C^*$. For functions in the class $\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the class $\mathcal B^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve …
