Statistical stability of equilibrium states for interval maps

TL;DR

This paper proves the continuous variation of equilibrium states and their densities for certain multimodal interval maps with polynomial derivative growth, especially near the parameter t=1, enhancing understanding of their statistical stability.

Contribution

It establishes the weak* continuity of equilibrium states and their densities for families of multimodal maps with polynomial growth, extending prior results to a broader class of maps.

Findings

Equilibrium states vary continuously in the weak* topology.

Densities of equilibrium states vary continuously when t=1.

Results apply to maps with polynomial growth of derivatives along critical orbits.

Abstract

We consider families of multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential $\phi_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. We show that these equilibrium states vary continuously in the weak$^*$ topology within such families. Moreover, in the case $t=1$, when the equilibrium states are absolutely continuous with respect to Lebesgue, we show that the densities vary continuously within these families.