Intermediate topological pressures and variational principles for nonautonomous dynamical systems

TL;DR

This paper introduces a family of intermediate topological pressures for nonautonomous dynamical systems, bridging existing pressure concepts and establishing their properties and variational principles.

Contribution

It defines a new parameterized pressure interpolating between known pressures, analyzes its properties, and connects it with measure-theoretic pressures via variational principles.

Findings

Intermediate pressure is continuous on (0,1] but may be discontinuous at 0.

The pressure satisfies the power rule and monotonicity.

Variational principles relate intermediate topological and measure-theoretic pressures.

Abstract

We introduce a one-parameter family of intermediate topological pressures for nonautonomous dynamical systems which interpolate between the Pesin-Pitskel topological pressure and the lower and upper capacity pressures. The construction is based on the Carath\'eodory-Pesin structure in which all admissible strings in a covering satisfy $ N \le n < N/\theta + 1 $, where $ \theta \in [0,1] $ is a parameter. The extremal cases $\theta=0$ and $\theta=1$ recover the Pesin-Pitskel pressure and the two capacity pressures, respectively. We first investigate several properties of the intermediate pressure, including proving that it is continuous on $(0, 1]$ but may fail to be continuous at $0$, as well as establishing the power rule and monotonicity. We then derive inequalities for intermediate pressures with respect to the factor map. Finally, we introduce intermediate measure-theoretic pressures and prove variational principles relating them to the corresponding topological pressures.