Modulus of continuity of averages of SRB measures for a transversal family of piecewise expanding unimodal maps

TL;DR

This paper investigates the regularity of the integral of a test function against SRB measures for a family of piecewise expanding unimodal maps, showing that under certain conditions, the regularity is precisely characterized and not Lipschitz.

Contribution

It establishes the non-Lipschitz nature and determines the exact modulus of continuity of the SRB measure averages for a transversal family of unimodal maps.

Findings

a) a0 ext{The function } \u03b3(t) ext{ is not Lipschitz for almost all } t.

a) ext{Provides the exact modulus of continuity of } b3(t).

a) ext{Identifies conditions under which regularity fails.}

Abstract

Let $f_t:[0,1] \to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t \in [a,b]$. If $\mu_t$ is the corresponding SRB measure for $f_t$, we study the regularity of $\Gamma(t)=\int \phi d\mu_t$ when assuming that the family is transversal to the topological classes of these maps, more precisely, we prove that if $J_t(c)=\sum_{k=0}^{\infty} \frac{v_t(f_t^k(c))}{Df_t^k(f_t(c))} \neq 0$ for all $t$, where $v_t(x)=\partial_s f_s(x)|_{s=t}$, then $\Gamma(t)$ is not Lipschitz for almost all $t\in [a,b]$. Furthermore, we give the exact modulus of continuity of $\Gamma(t)$.