Invariant measures for interval maps without Lyapunov exponents

TL;DR

This paper constructs an invariant measure for a complex interval map with a Lorenz-like singularity, where the Lyapunov exponent is not defined, highlighting unusual dynamical behavior.

Contribution

It introduces a new invariant measure for a class of interval maps lacking a well-defined Lyapunov exponent, expanding understanding of non-uniform hyperbolic systems.

Findings

Invariant measure exists despite undefined Lyapunov exponents.

Set of full measure where pointwise Lyapunov exponent is undefined.

Map features Lorenz-like singularity and non-flat critical points.

Abstract

We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.