High pointwise emergence and Katok's conjecture for systems with non-uniform structure

TL;DR

This paper investigates the set of points with high pointwise emergence in non-uniform dynamical systems, proving it has full topological pressure and establishing the existence of ergodic measures with arbitrary intermediate pressures.

Contribution

It introduces the study of high pointwise emergence in non-uniform systems and proves these sets have full topological pressure, advancing understanding of their complex behavior.

Findings

High pointwise emergence sets have full topological pressure.

Existence of ergodic measures with arbitrary intermediate pressures.

Extension of pointwise emergence concepts to non-uniform systems.

Abstract

Recently, Kiriki, Nakano and Soma introduced a concept called pointwise emergence as a new quantitative perspective into the study of non-existence of averages for dynamical systems. In the present paper, we consider the set of points with high pointwise emergence for systems with non-uniform structure and prove that this set carries full topological pressure. For the proof of this result, we show that such systems have ergodic measures of arbitrary intermediate pressures.