Typical properties of interval maps preserving the Lebesgue measure

TL;DR

This paper studies the typical properties of continuous interval maps that preserve Lebesgue measure, analyzing their dynamical behavior within a complete metric space of such maps.

Contribution

It characterizes the generic dynamical properties of measure-preserving continuous maps on the interval in the space endowed with the uniform metric.

Findings

Identifies typical dynamical behaviors of measure-preserving maps

Provides a classification of generic properties in the space

Enhances understanding of measure-preserving interval dynamics

Abstract

Let us denote $\lambda$ the Lebesgue measure on $[0,1]$, put$$ C(\lambda)=\{f\in C([0,1]);\ \forall~A\subset [0,1], A~\text{Borel}:\ \lambda(A)=\lambda(f^{-1}(A))\}.$$ We endow the set $C(\lambda)$ by the uniform metric $\rho$ and investigate dynamical properties of typical maps in the complete metric space $(C(\lambda),\rho)$.