Structure and f-dependence of the a.c.i.m. for a unimodal map f of Misiurewicz type

TL;DR

This paper investigates the ergodic properties of Misiurewicz-type unimodal maps, revealing the structure of their invariant measures as a sum of singular spikes and continuous parts, and discusses differentiability of the invariant measure with respect to the map.

Contribution

It provides a detailed analysis of the invariant measure's structure for Misiurewicz unimodal maps using transfer operators on a Banach space, highlighting the measure's spike and background decomposition.

Findings

Invariant measure decomposes into 1/√spikes plus continuous background

Explicit description of measure's structure along critical orbit

Discussion on differentiability of the map to measure relationship

Abstract

By using a suitable Banach space on which we let the transfer operator act, we make a detailed study of the ergodic theory of a unimodal map $f$ of the interval in the Misiurewicz case. We show in particular that the absolutely continuous invariant measure $\rho$ can be written as the sum of 1/square root spikes along the critical orbit, plus a continuous background. We conclude by a discussion of the sense in which the map $f\mapsto\rho$ may be differentiable.