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Regul. Chaotic Dyn., 2025 Volume 30, Issue 1, Pages 93–102 (Mi rcd1298)

On the Existence of Expanding Attractors with Different Dimensions

Vladislav S. Medvedev, Evgeny V. Zhuzhoma

National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, 603005 Nizhny Novgorod, Russia

Abstract: We prove that an $n$-sphere $\mathbb{S}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with nonorientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{2}]\}$ where $[x]$ is the integer part of $x$. In addition, any $n$-sphere $\mathbb{S}^n$, $n\geqslant 3$, admits axiom A diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{3}]\}$. We prove that an $n$-torus $\mathbb{T}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{T}^n\to\mathbb{T}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,n-1\}$. We also prove that, given any closed $n$-manifold $M^n$, $n\geqslant 2$, and any $d\in\{1,\ldots,[\frac{n}{2}]\}$, there is an axiom A diffeomorphism $f: M^n\to M^n$ with a $d$-dimensional nonorientable expanding attractor. Similar statements hold for axiom A flows.

Keywords: axiom A systems, basic set, expanding attractor

MSC: 58C30, 37D15

Received: 26.07.2024
Accepted: 22.11.2024

Language: English

DOI: 10.1134/S1560354724580020



© Steklov Math. Inst. of RAS, 2025