Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem

TL;DR

This paper investigates the spectral gap property and phase transition phenomena for positively expansive dynamical systems and smooth potentials, extending understanding from one-dimensional cases to higher dimensions.

Contribution

It demonstrates the spectral gap property for transfer operators in positively expansive maps and identifies phase transition behaviors in certain high-dimensional systems.

Findings

Transfer operator has spectral gap for a broad class of potentials in positively expansive maps.

Phase transition phenomena are observed in specific high-dimensional skew-product systems.

Extends previous results from circle maps to higher-dimensional dynamics.

Abstract

It is known that all uniformly expanding dynamics $f: M \rightarrow M$ have no phase transition with respect to a H\"older continuous potential $\phi : M \rightarrow \mathbb{R}$, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f , t\phi)$ is analytical. Moreover, the associated transfer operator $\mathcal{L}_{f , t\phi}$, acting on the space of H\"older continuous functions, has the spectral gap property for $t \in \mathbb{R}$. For dynamics that are topologically conjugate to an expanding map, a full understanding has yet to be achieved. On the one hand, by \cite{KQW21,KQ22}, for such maps and continuous potentials, the associated topological pressure function can behave wildly. On the other hand, by \cite{BF23}, for transitive local diffeomorphisms on the circle and a large class of H\"older continuous potentials, the phase transition does not occur, and the associated transfer operator has the spectral gap property for all parameters $t \in \mathbb{R}$. As a first approach to understanding what happens in high dimensions, in this paper, we study positively expansive local diffeomorphisms. In particular, we show that the associated transfer operator has the spectral gap property for a large class of regular potentials. Moreover, for a class of intermittent skew-products and a large class of regular potentials, we obtain phase transition results analogous to \cite{BF23}.