Invariant measures for interval maps with critical points and singularities

TL;DR

This paper proves that certain interval maps with critical points and singularities have ergodic invariant measures absolutely continuous with respect to Lebesgue measure, under mild growth conditions.

Contribution

It establishes the existence of absolutely continuous invariant measures for a broad class of interval maps with critical points and singularities, extending previous results.

Findings

Existence of ergodic absolutely continuous invariant measures under mild conditions

Applicable to maps with discontinuities and infinite derivatives

Extends theory to cusp maps with singularities

Abstract

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.