Search

We study the global topology of the space $\mathcal L$ of loops of contactomorphisms of a non-orderable closed contact manifold $(M^{2n+1}, \alpha)$. We filter $\mathcal L$ by a quantitative measure of the ``positivity'' of the loops and describe the topology of $\mathcal L$ in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of $\mathcal L$ are subgroups of the homotopy groups of the subspace of positive loops $\mathcal L^+$. We obtain analogous results for the space of loops of Legendrian submanifolds in $(M^{2n+1}, \alpha)$., Comment: 11 pages, 4 figures

Let $S^m = \{x_0^2 + x_1^2 + \cdots + x_m^2 = 1\}$ and $P = \{x_0 = x_1 = 0\} \cap S^m$. Suppose that $f$ is a self--map of $S^m$ such that $f^{-1}(P) = P$ and $|\mathrm{deg}(f_{|P})| < |\mathrm{deg}(f)|$. Then, the number of fixed points of $f^n$ grows at least exponentially with base $|d| > 1$, where $d = \mathrm{deg}(f)/\mathrm{deg}(f_{|P}) \in \mathbb Z$., Comment: 10 pages, 1 figure

The evaluation of Alexander-Spanier cochains over formal simplices in a topological space leads to a notion of integration of Alexander-Spanier cohomology classes over \v{C}ech homology classes. The integral defines an explicit and non-degenerate pairing between the Alexander-Spanier cohomology and the \v{C}ech homology. Instead of working on the limits that define both groups, most of the discussion is carried out "at scale $\mathcal U$", for an open covering $\mathcal U$. As an application, we generalize a result of Manning to arbitrary compact spaces $X$: we prove that the topological entropy of $f \colon X \to X$ is bounded from below by the logarithm of the spectral radius of the map induced in the first \v{C}ech cohomology group.

We construct two planar homeomorphisms $f$ and $g$ for which the origin is a globally asymptotically stable fixed point whereas for $f \circ g$ and $g \circ f$ the origin is a global repeller. Furthermore, the origin remains a global repeller for the iterated function system generated by $f$ and $g$ where each of the maps appears with a certain probability. This planar construction is also extended to any dimension greater than 2 and proves for first time the appearance of the Parrondo's dynamical paradox in odd dimensions.

Let $X$ be a compact, metric and totally disconnected space and let $f:X\to X$ be a continuos map. We relate the eigenvalues of $f_{*}:\check{H}_{0}(X;\mathbb{C})\to\check{H}_{0}(X;\mathbb{C})$ to dynamical properties of $f$, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0,1 is an eigenvalue. This stands in contrast with the classical Manning's inequality., Comment: Version accepted for publication in Ergod. Theory Dyn. Syst

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of $C^0$--small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary., Comment: 9 pages, 1 figure

In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to \mathbb{R}^d$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. Our approach exploits certain attractor-repeller decomposition of the unstable manifold, reducing the study of the cohomological Conley index to the relation between the cohomology of an attractor and its basin of attraction. This is a classical problem that, in the present case, is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated. To address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology: if $Z$ is a metric space and $K \subseteq Z$ is a compact, global attractor for a continuous map $g \colon Z \to Z$, we show how to interpret series of the form $\sum_{j \ge 0} a_j (g^*)^j$ as endomorphisms of the cohomology group of the pair $(Z,K)$., Comment: 55 pages

Let $X$ be a closed invariant subset of the half--open annulus $\mathbb{A}$ such that $\mathbb{A} \setminus X$ is homeomorphic to $\mathbb{A}$. We prove that either the rotation number of all forward semi--orbits of accessible points of $X$ are well--defined and equal to the prime end rotation number or the same is true for all backward semi--orbits of accessible points of $X$., Comment: 35 pages, 10 figures. To appear in J. Lond. Math Soc. (2)

We prove the existence of minimal symplectomorphisms and strictly ergodic contactomorphisms on manifolds which admit a locally free $\mathbb{S}^1$--action by symplectomorphisms and contactomorphisms, respectively. The proof adapts the conjugation method, introduced by Anosov and Katok, to the contact and symplectic setting.

Matsumoto proved in arXiv:1012.0981 that the prime end rotation numbers associated to an invariant annular continuum are contained in its rotation set. An alternative proof of this fact using only simple planar topology is presented., Comment: 7 pages, 2 figures. Final version to appear in Nagoya Math. J

We prove that for every integer sequence $I$ satisfying Dold relations there exists a map $f : \mathbb{R}^d \to \mathbb{R}^d$, $d \ge 2$, such that $\mathrm{Per(f)} = \mathrm{Fix(f)} = \{o\}$, where $o$ denotes the origin, and $(i(f^n, o))_n = I$., Comment: 11 pages, 2 figures. Final version to appear in Topol. Methods Nonlinear Anal

We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set and continuous maps in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 1$. This characterization allows us to compute effectively the Lefschetz zeta functions for a wide class of continuous maps in the 2-sphere, to obtain new results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of Shub about the growth of the number of periodic orbits of degree--$d$ maps in the 2-sphere., Comment: 15 pages, 4 figures. Final version published in DCDS

A characterization of dynamically defined zeta functions is presented. It comprises a list of axioms, natural extension of the one which characterizes topological degree, and a uniqueness theorem. Lefschetz zeta function is the main (and proved unique) example of such zeta functions. Another interpretation of this function arises from the notion of symmetric product from which some corollaries and applications are obtained., Comment: 25 pages, 2 figures. Final version to appear in J. Fixed Point Theory Appl

This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a local homeomorphism of $\mathds{R}^d$ and an invariant and isolated acyclic continuum, such as a cellular set or a fixed point. In this setting, we obtain a complete description of the first discrete homological Conley index, which is periodic, that enforces a combinatorial behavior of higher indices. As a consequence, we prove that isolated (as an invariant set) fixed points of orientation-reversing homeomorphisms of $\mathds{R}^3$ have fixed point index $\le 1$ and, as a corollary, that there are no minimal orientation-reversing homeomorphisms in $\mathds{R}^3$., Comment: 35 pages