Hausdorff dimension of Kuperberg minimal sets

TL;DR

This paper investigates the Hausdorff dimension of Kuperberg minimal sets by linking their symbolic dynamics to graph directed pseudo-Markov systems, providing new insights into their fractal structure.

Contribution

It introduces novel techniques to analyze the dimension theory of Kuperberg minimal sets through symbolic dynamics and graph directed systems.

Findings

Established a connection between Kuperberg minimal sets and pseudo-Markov systems.

Derived new estimates for the Hausdorff dimension of these minimal sets.

Enhanced understanding of the fractal geometry of minimal sets in smooth flows.

Abstract

In 1994, Kuperberg constructed a smooth flow on a three-manifold with no periodic orbits. It was later shown that a generic Kuperberg flow preserves a codimension one laminar minimal set. We develop new techniques to study the symbolic dynamics and dimension theory of this minimal set, by relating it to the limit set of a graph directed pseudo-Markov system over a countable alphabet.