Geometric Structure and Ergodic Properties of Bony Multi-Graphs

TL;DR

This paper investigates the geometric structure of invariant multi-graphs in skew product systems, introducing bony multi-graphs and analyzing their ergodic and thermodynamic properties, with extensions to baker map systems.

Contribution

It introduces invariant bony multi-graphs and constructs skew products with attracting multi-graphs supporting finitely many ergodic SRB measures.

Findings

Existence of attracting invariant multi-graphs and bony multi-graphs.

Construction of skew products with specific ergodic properties.

Extension of results to skew products over generalized baker maps.

Abstract

The main goal in this paper is to describe the geometric structure of invariant graphs of a certain class of skew products. Our focus is on attracting multi-graphs. An invariant multi-graph is an invariant compact set which is a finite union of invariant graphs, and thus consists of a finite number of points on each fiber. We introduce invariant bony multi-graphs and construct an open set of skew products over an invertible base map (solenoid map) having attracting invariant multi-graphs and bony multi-graphs which support finitely many ergodic SRB measures. In this study some thermodynamic properties are investigated. Finally, we extend our results to a family of skew products over a generalized baker map.