New exotic minimal sets from pseudo-suspensions of Cantor systems

TL;DR

This paper introduces pseudo-suspension techniques to construct exotic minimal sets in dynamical systems, providing new examples of hereditarily indecomposable continua with complex dynamics and minimal sets in smooth 4-manifolds.

Contribution

It develops a novel pseudo-suspension method for Cantor systems, producing hereditarily indecomposable continua with intermediate complexity and minimal sets with diverse dynamical properties.

Findings

Constructed hereditarily indecomposable continua with intermediate complexity

First examples of minimal, uniformly rigid, and weakly mixing homeomorphisms in dimension 1

Realized these examples as invariant sets of smooth 4-manifold diffeomorphisms

Abstract

We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel-Anosov-Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that admits a homeomorphism that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms of intermediate complexity. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds. We also use our techniques to exhibit the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension $1$, and these can also be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least $2$.