Related papers: Tame dynamics and robust transitivity
In the theory of renormalization for classical dynamical systems, e.g. unimodal maps and critical circle maps, topological conjugacy classes are stable manifolds of renormalization. Physically more realistic systems on the other hand may…
Given a dynamical system $(X,T)$ and a family $\mathsf{I}\subseteq \mathcal{P}(\omega)$ of "small" sets of nonnegative integers, a point $x \in X$ is said to be $\mathsf{I}$-strong universal if for each $y \in X$ there exists a subsequence…
We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become…
In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the homoclinic classes is not…
We describe an example of a $C^\infty$ diffeomorphism on a 7--manifold which has a compact invariant set such that uncountably many of its connected components are pseudocircles. (Any 7--manifold will suffice.) Furthermore, any…
An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same…
The aim of this paper is twofold. First, we introduce standard blenders (special hyperbolic sets) and their variations, and prove their fundamental properties on the generation of $C^1$-robust tangencies. In particular, these blenders…
Homoclinic classes of generic $C^1$-diffeomorphisms are maximal transitive sets and pairwise disjoint. We here present a model explaining how two different homoclinic classes may intersect, failing to be disjoint. For that we construct a…
In this paper we consider closed orientable surfaces $S$ of positive genus and $C^r$-diffeomorphisms $f:S\rightarrow S$ isotopic to the identity ($r\geq 1)$. The main objective is to study periodic open topological disks which are…
In generic dynamical systems heteroclinic cycles are invariant sets of codimension at least one, but they can be structurally stable in systems which are equivariant under the action of a symmetry group, due to the existence of…
We consider the concept of strong pluripotency of dynamical systems for a hyperbolic invariant set, as introduced in [KNS]. To the best of our knowledge, for the whole hyperbolic invariant set, the existence of robust strongly pluripotent…
The last years have witnessed rapid progress in the topological characterization of out-of-equilibrium systems. We report on robust signatures of a new type of topology -- the Euler class -- in such a dynamical setting. The enigmatic…
Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations…
A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \N$, or it is a "tame" topological space whose topology…
We study a certain class circle maps which are constant on one interval (called flat piece), and such that the degrees of the singularities at the boundary of the flat piece are different. In this paper, we show that if the topological…
We prove that various classical conformal diffeomorphism groups, which are known to be essential [1], are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological…
In this paper, we give an answer to a $C^{r}$ $(2\leq r <\infty)$ version of the open problem of Takens in [Nonlinearity, 21 (2008), no.3, T33-T36] which is related to historic behavior of dynamical systems. To obtain the answer, we show…
In this paper we study the general concept of integrability in the broad sense within the frame of differential Galois theory. We concentrate on the gradient systems which are not integrable. In spite of it, if we consider them as the real…
We present new examples of open sets of diffeomorphisms such that a generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that…
Given any triplet of positive integers $n \geq 2$, $m$ and $k$ such that $n=m+k$, we exhibit a $C^1$ robustly transitive endomorphism of $\mathbb{T}^n$ with persistent critical points in the isotopy class of $F \times Id$, where $F$ is an…