Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1903.11644

Kneading sequences for toy models of Hénon maps

Ermerson Araujo
View original source ↗Read source document ↗Export metadata ↓

Abstract

The purpose of this article is to study the relation between combinatorial equivalence and topological conjugacy, specifically how a certain type of combinatorial equivalence implies topological conjugacy. We introduce the concept of kneading sequences for a setting that is more general than one-dimensional dynamics: for the two-dimensional toy model family of Hénon maps introduced by Benedicks and Carleson, we define kneading sequences for their critical lines, and prove that these sequences are a complete invariant for a natural conjugacy class among the toy model family. We also establish a version of Singer's Theorem for the toy model family.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ermerson Araujo. 2019-03-27. Kneading sequences for toy models of Hénon maps. https://doi.org/10.1017/etds.2020.117

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

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations

In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that no admissible choice of weighting function can, in general, overcome the lower bounds imposed by this regularity. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.

math.DS↗
arXiv2026

Spectral theory of frame flows on closed hyperbolic manifolds

We prove a resolvent estimate for the generator of the frame flow on hyperbolic manifolds away from vertical lines of resonances. A byproduct of the proof is an optimal essential spectral gap property for the generator, hence giving another proof of exponential mixing of frame flows with respect to the volume measure of the frame bundle. This extends the result of [https://arxiv.org/abs/2005.08387v2] in dimension 3 to any dimension. We make extensive use of the Borel-Weil calculus developed in [https://arxiv.org/abs/2405.14846] to overcome difficulties of this higher-dimensional case.

math.DS↗