Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1410.1758

Steps towards a classification of $C^r$-generic dynamics close to homoclinic points

Nicolas Gourmelon
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present here the first part of a program for a classification of the generic dynamics close to homoclinic and heteroclinic points, in the $C^r$ topologies, $r\geq 1$. This paper only contains announcements and a few sketches of proofs; a forthcoming series of papers will present the proofs in details. The two prototypical examples of non-hyperbolic dynamics are homoclinic tangencies and heterodimensional cycles. Palis conjectured that they actually characterize densely non-hyperbolic dynamics. It is therefore important to understand what happens close to those bifurcations. We generalize classical results of Newhouse, Palis and Viana, for both tangencies and cycles: close to a homoclinic tangency or to a heterodimensional cycle there is abundance of diffeomorphisms exhibiting infinitely many sinks or sources if and only if the dynamics is not volume-hyperbolic. This proves in particular a conjecture of Turaev for homoclinic tangencies. An important result of Bonatti, Diaz, Pujals states that if a homoclinic class is $C^1$-robustly without dominated splitting, then nearby diffeomorphisms exhibit $C^1$-generically infinitely many sinks or sources. We show that this holds in higher regularities, under the further assumption that non-dominations are obtained through so-called "mechanisms". This includes all the examples of robustly non-dominated homoclinic classes one can build with the tools known up to now. We actually have a $C^r$-equivalent of a recent $C^1$-result of Bochi and Bonatti: we describe precisely the Lyapunov exponents along periodic points that may appear close to a homoclinic tangency or to a homoclinic class. The results of Newhouse, Palis and Viana were proven for the $C^r$ topologies, $r\geq 2$. Our results hold also in the $C^{1+α}$ topologies.

Explore related subjects

math.DS

Keep this discovery

BibTeXRIS

Nicolas Gourmelon. 2014-10-29. Steps towards a classification of $C^r$-generic dynamics close to homoclinic points. https://arxiv.org/abs/1410.1758

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗