arXiv 1401.02322014-01math.DS

On the Finiteness of Attractors for One-Dimensional Maps with Discontinuities

Paulo Brandão, Jacob Palis, Vilton Pinheiro

备注：: 33 pages, 8 figures

英文摘要：

Since the proof, at the end of the 80's, of the finiteness of the number of attractors for $C^3$ maps of the interval having negative Schwarzian derivative, it has been generally considered that the same result could be true for maps with discontinuities. In the present paper we show that this is indeed the case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Finiteness of attractors and repellers on sectional hyperbolic sets

We obtain an upper bound for the number of attractors and repellers that can appear from small perturbations of a sectional hyperbolic set. This extends results from [Sectional-Anosov flows in higher dimensions] and [The explosion of singular-hyperbolic attractors].

2013-09-22math.DS

On the Finiteness of Attractors for piecewise $C^2$ Maps of the Interval

We consider piecewise $C^2$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^2$ non-flat map of the interval displays only a finite number of non-periodic attractors.

2016-03-11math.DS

Topological and statistical attractors for interval maps

We use the concept of Baire Ergodicity and Ergodic Formalism introduced to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the {\em wandering intervals attractors}. As a result, we establish the finiteness of the non-periodic topological attractors for piecewise $C^2$ maps with discontinuities. For $C^2$ interval maps without discontinuities, we show the coincidence of the statistical attractors with the topological ones and we calculate the upper Birkhoff averages of continuous functions for generic points, even when the map …

2022-02-03math.DS

Finiteness of partially hyperbolic attractors with one-dimensional center

We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-generic diffeomorphisms far from homoclinic tangencies in dimension $3$ either have at most finitely many attractors, or satisfy Newhouse phenomenon.

2019-12-10math.DS

Emergence and non-typicality of the finiteness of the attractors in many topologies

We will introduce the notion of Emergence for a dynamical system, and we will conjecture the local typicality of super-polynomial ones. Then, as part of this program, we will provide sufficient conditions for an open set of Cd-families of Cr-dynamics to contain a Baire generic set formed by families displaying infinitely many sinks at every parameter, for all 1 \le d \le r with d finite (and r possibly infinite) for two different topologies on families. In particular the case d=r=1 is new.

2017-03-10math.DS

Finiteness of homoclinic classes on sectional hyperbolic sets

We study small perturbations of a sectional hyperbolic set of a vector field on a compact manifold. Indeed, we obtain robustly finiteness of homoclinic classes on this scenary. Moreover, since attractor and repeller sets are particular cases of homoclinic classes, this result improve (A. M. López B, Finiteness and existence of attractors and repellers on sectional hyperbolic sets, Discrete and Continuous Dynamical Systems-A 37).

2019-08-12math.DS

更多相关论文收起更多论文

Statistical properties of interval maps with critical points and discontinuities

We consider dynamical systems given by interval maps with a finite number of turning points (including critical points, discontinuities) possibly of different critical orders from two sides. If such a map $f$ is continuous and piecewise $C^2$, satisfying negative Schwarzian derivative and some summability conditions on the growth of derivatives and recurrence along the turning orbits, then $f$ has finitely many attractors whose union of basins of attraction has total probability, and each attractor supports an absolutely continuous invariant probability measure $μ$. Over each attractor there e…

2010-01-11math.DS

Structure of attractors for boundary maps associated to Fuchsian groups

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 …

2017-05-03math.DS

拓扑吸引子与历史行为的遍历形式体系

Ergodic Formalism for topological Attractors and historic behavior

本文引入 Baire 遍历性和遍历形式体系研究拓扑与统计吸引子，确立了其存在性与有限性，并应用于多种映射与系统，为具有丰富历史行为的动力系统中一般轨道的研究提供了新方法。

2024-05-31math.DS

On the Asymptotic Properties of Piecewise Contracting Maps

We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes…

2014-04-01math.DS

全局吸引子猜想在特殊情况下的证明

A Proof of the Global Attractor Conjecture in a Special Case

University of Copenhagen（哥本哈根大学）

本文在特定结构条件下证明了复平衡质量作用反应网络的全局吸引子猜想，通过两步证明确保正轨迹收敛到唯一正平衡点。

2026-09-22math.DSq-bio.MN