Super-expanding measures

TL;DR

This paper investigates the dynamics of one-dimensional Lorenz maps, revealing conditions under which they exhibit uncountably many ergodic invariant measures with infinite Lyapunov exponents and positive entropy, depending on the recurrence rate of the singularity.

Contribution

It establishes the existence of a dense subset of Lorenz maps with complex invariant measures and characterizes the influence of singularity recurrence on Lyapunov exponents.

Findings

Dense subset of Lorenz maps with uncountable ergodic measures with infinite Lyapunov exponents.

Fast recurrence to the singularity leads to measures with positive entropy.

Slow recurrence bounds the Lyapunov exponents of all invariant measures.

Abstract

We study the one-dimensional expanding Lorenz maps and show the existence of dense subset D of Lorens maps such that each f in D has an uncountable set of ergodic invariant probabilities with infinite Lyapunov exponent and positive entropy. Such measures may appear when the singularity has fast recurrence to itself. Conversely, if the singularity has slow recurrence to itself then the Lorenz map has an upper bound to the Lyapunov exponent of all invariant measures.