On existence of expanding attractors with different dimensions

TL;DR

This paper constructs structurally stable diffeomorphisms with non-orientable and orientable expanding attractors of various dimensions on spheres, tori, and general closed manifolds, advancing understanding of dynamical systems' attractor structures.

Contribution

It demonstrates the existence of structurally stable diffeomorphisms with expanding attractors of arbitrary dimension and orientability on standard manifolds and general closed manifolds.

Findings

Existence of non-orientable expanding attractors on spheres for various dimensions.

Existence of orientable expanding attractors on tori for various dimensions.

Construction of axiom A diffeomorphisms with specified attractors on general manifolds.

Abstract

We prove that $n$-sphere $\mathbb{S}^n$, $n\geq 2$, admits structurally stable diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with non-orientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{2}]\}$ where $[x]$ is an integer part of $x$. One proves that $n$-torus $\mathbb{T}^n$, $n\geq 2$, admits structurally stable diffeomorphisms $\mathbb{T}^n\to\mathbb{T}^n$ with orientable expanding attractors of any topological dimension $1\leq q\leq n-1$. We also prove that given any closed $n$-manifold $M^n$, $n\geq 2$, 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 non-orientable expanding attractor. Similar statements hold for axiom A flows.