Ergodic theory for smooth one-dimensional dynamical systems

TL;DR

This paper explores the measurable dynamics of smooth one-dimensional maps, detailing decompositions of attractors and ergodic components, extending previous work to more general smooth cases.

Contribution

It introduces a new approach to analyze the ergodic theory of smooth one-dimensional maps beyond those with negative Schwarzian derivative.

Findings

Decomposition of global measure-theoretical attractor into primitive components

Ergodic decomposition of the system

Hopf decomposition for smooth maps

Abstract

In this paper we study measurable dynamics for the widest reasonable class of smooth one dimensional maps. Three principle decompositions are described in this class : decomposition of the global measure-theoretical attractor into primitive ones, ergodic decomposition and Hopf decomposition. For maps with negative Schwarzian derivative this was done in the series of papers [BL1-BL5], but the approach to the general smooth case must be different.