A countable partition for singular flows, and its application on the entropy theory

TL;DR

This paper introduces a novel countable partition for hyperbolic singular flows, facilitating the analysis of entropy properties and demonstrating upper semi-continuity of the entropy function.

Contribution

It constructs a new partition and cross section at singularities, enabling entropy analysis for singular flows away from homoclinic tangencies.

Findings

Partition has finite metric entropy for all invariant measures

Elements of the partition stay in scaled neighborhoods for long times

Entropy function is upper semi-continuous with respect to measures and flows

Abstract

In this paper, we construct a countable partition $\mathscr{A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $\mathscr{A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.