Variation of Physical Measures in Nontrivial Mixed Partially Hyperbolic Systems

TL;DR

This paper constructs a smooth mixed partially hyperbolic system, identifies its skeleton, and demonstrates how perturbations can cause the number of physical measures to vary, advancing understanding of such dynamical systems.

Contribution

It provides explicit examples of mixed partially hyperbolic systems with variable physical measures and analyzes their stability under perturbations.

Findings

Support of physical measures contains three fixed points with distinct unstable indices

Number of physical measures varies upper semi-continuously under perturbations

Constructs explicit examples sharing characteristics with classical systems

Abstract

We construct a smooth nontrivial mixed partially hyperbolic system and explicitly identify its skeleton. This example shares characteristics with the classical examples. Moreover, the support of each physical measure contains three fixed points with mutually distinct unstable indices. By appropriately perturbing the skeleton, we provide an example where the number of physical measures varies upper semi-continuously. The general framework of mixed partially hyperbolic systems has been studied in theorem.