Intermediate topological pressures and variational principles for nonautonomous dynamical systems
Yujun Ju
TL;DR
The work introduces a $\theta$-parameterized interpolation of topological pressures for nonautonomous dynamical systems, connecting the Pesin–Pitskel construction ($\theta=0$) with capacity pressures ($\theta=1$) via a Carathéodory–Pesin framework that restricts admissible string lengths. It establishes fundamental properties, including continuity in $\theta$ on $(0,1]$, a power rule, and factor-map inequalities, and provides an equivalent Bowen-ball formulation. The paper further develops a variational principle by defining $\theta$-intermediate measure-theoretic pressures and proving that topological pressures equal the supremum of their measure-theoretic counterparts over invariant measures supported on the relevant sets. This unifies several nonautonomous dynamical invariants and yields a refined toolkit for analyzing dynamical complexity on time-varying or noncompact spaces, with the two endpoint cases recovering known pressures. The results have potential implications for fine-grained entropy and pressure analyses in nonautonomous and noncompact dynamics, offering a continuum of pressures that interpolate between established theories.
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 Carathéodory-Pesin structure in which all admissible strings in a covering satisfy $ N \le n < N/θ+ 1 $, where $ θ\in [0,1] $ is a parameter. The extremal cases $θ=0$ and $θ=1$ 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.