A Variational Principle for the Topological Pressure of Non-autonomous Iterated Function Systems on Subsets
Yujun Ju, Lingbing Yang
TL;DR
The paper develops a Pesin–Pitskel style topological pressure for non-autonomous iterated function systems on subsets by employing the Carathéodory–Pesin structure. It proves that this pressure coincides with a weighted topological pressure and establishes a variational principle stating that, for any nonempty compact set $Z$, the topological pressure $P_Z(\boldsymbol{f},\varphi)$ equals the supremum of measure-theoretic pressures $P_\mu(\boldsymbol{f},\varphi)$ over all Borel probability measures supported on $Z$. The main methodology combines the CP construction with Vitali covering arguments to connect dynamical and measure-theoretic quantities, extending classical autonomous and semigroup theories to NAIFSs. These results provide a robust thermodynamic formalism for non-autonomous and semigroup-driven dynamics on subsets, enabling further multifractal and dimension-theoretic analyses in this generalized setting.
Abstract
Motivated by the notion of topological entropy for free semigroup actions introduced by Biś, we define the Pesin--Pitskel topological pressure for non-autonomous iterated function systems via the Carathéodory--Pesin structure. We show that this Pesin--Pitskel topological pressure coincides with the corresponding weighted topological pressure. Furthermore, we establish a variational principle asserting that, for any nonempty compact subset, the Pesin--Pitskel topological pressure equals the supremum of the associated measure-theoretic pressures over all Borel probability measures supported on that subset.