Heterodimensional cycles of hyperbolic ergodic measures

TL;DR

This paper introduces and studies rich heterodimensional cycles of hyperbolic ergodic measures, showing their relation to periodic measures and constructing examples with uncountably many related measures.

Contribution

It defines rich heterodimensional cycles of measures and establishes their properties within partially hyperbolic systems and skew products.

Findings

Measures linked by rich heterodimensional cycles form segments in the measure space.

Such cycles are dense in the closure of measures supported on periodic orbits.

Constructs examples with uncountably many measures related by these cycles.

Abstract

We introduce the concept of a heterodimensional cycle of hyperbolic ergodic measures and a special type of them that we call rich. Within a partially hyperbolic context, we prove that if two measures are related by a rich heterodimensional cycle, then the entire segment of probability measures linking them lies within the closure of measures supported on periodic orbits. Motivated by the occurrence of robust heterodimensional cycles of hyperbolic basic sets, we study robust rich heterodimensional cycles of measures providing a framework for this phenomenon for diffeomorphisms. In the setting of skew products, we construct an open set of maps having uncountably many measures related by rich heterodimensional cycles.