Wandering intervals and absolutely continuous invariant probability measures of interval maps

TL;DR

This paper establishes conditions under which certain interval maps with critical points and discontinuities have no wandering intervals and possess absolutely continuous invariant measures, extending previous methods to more complex maps.

Contribution

It introduces new summability conditions that guarantee the existence of absolutely continuous invariant measures for piecewise $C^1$ interval maps with critical points and discontinuities.

Findings

No wandering intervals under the given conditions

Existence of absolutely continuous invariant measures

Bounded backward contraction property established

Abstract

For piecewise $C^1$ interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recurrence along critical orbits, we prove the nonexistence of wandering intervals, the existence of absolutely continuous invariant measures, and the bounded backward contraction property. The proofs are based on the method of proving the existence of absolutely continuous invariant measures of unimodal map, developed by Nowicki and van Strien.