Hyperbolic absolutely continuous invariant measures for C^r one-dimensional maps

TL;DR

This paper demonstrates that certain smooth one-dimensional maps with positive Lyapunov exponents support hyperbolic absolutely continuous invariant measures, with their basins covering almost all points where the exponent condition holds.

Contribution

It establishes the existence of hyperbolic absolutely continuous invariant measures for C^r maps on the interval or circle under specific Lyapunov exponent conditions, extending previous results.

Findings

Existence of hyperbolic ACIMs under Lyapunov exponent conditions

Basins of these measures cover the set of points with positive Lyapunov exponent almost everywhere

Application of Ledrappier-Young entropy characterization to non-invertible maps

Abstract

For r > 1, we show, using the Ledrappier-Young entropy characterization of SRB measures for non-invertible maps, that if a C^r map f of the interval or the circle has its Lyapunov exponent greater than 1/r log ||f ' || $\infty$ on a set E of positive Lebesgue measure, then it admits hyperbolic ergodic invariant measures that are absolutely continuous with respect to the Lebesgue measure. We also show that the basins of these measures cover E Lebesgue-almost everywhere.