Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e.

TL;DR

The paper investigates generic Lebesgue measure-preserving interval maps and shows that, within the space $C_\lambda$ of such continuous maps on $I=[0,1]$, the set of zero measure-theoretic entropy maps is dense and forms a $G_\delta$ subset, while topological entropy can be large or infinite. It develops a dyadic-partition framework to characterize entropy, constructs explicit zero-entropy examples, and proves that generically these maps are of nonmonotonic type, with normal level sets. The results reveal a striking divergence between topological and measure-theoretic dynamics for typical maps and have implications for planar attractors via inverse limits, where pseudo-arc attractors arise yet the dynamics on the attractor remains measure-theoretically simple (zero entropy) and weakly mixing. Together, the findings enhance understanding of entropy, invertibility a.e., and the intricate relationship between one-dimensional dynamics and higher-dimensional attractors.

Abstract

We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this article we show that thegeneric map has zero measure-theoreticentropy. This implies that there are dramatic differences inthe topological versus measure-theoretic behavior both for injectivity as well as for the structure of thelevel sets of generic maps. As a consequence we get a surprising corollary for a family of planar attractors homeomorphic to the pseudo-arcs.

Figures (2)

• Figure 1: Removing irrational critical values.
• Figure 2: Left: the map $\tilde{f}$ in step II. Right: the map $F$ in step III.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Corollary 3
• proof
• Remark 4
• Theorem
• Lemma 5
• proof
• Lemma 6
• proof