A Variational Principle for the Topological Pressure of Non-autonomous Iterated Function Systems on Subsets

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.

💡 Research Summary

The paper addresses a gap in the thermodynamic formalism for non‑autonomous iterated function systems (NAIFS), which are dynamical systems generated by a sequence of finite families of continuous maps that may vary with time. While previous works on NAIFS and related structures (free semigroup actions, non‑autonomous dynamical systems) have focused on “averaged” notions of complexity—most notably the Bufetov topological entropy—the maximal (or “Biś”) entropy, which captures the supremum of orbit growth rates, has not been systematically incorporated into a pressure theory.

The authors first recall the Biś topological entropy for free semigroup actions, defined via the growth rate of maximal (n,ε)-separated sets with respect to the metric d_max^n that simultaneously monitors all compositions of length n. They then embed NAIFS into this framework by constructing an equicontinuous metric d_*^n that coincides with d_max^n when the NAIFS reduces to a free semigroup action, thereby establishing that the topological sup‑entropy H(f;Y) of an NAIFS equals the Biś entropy h(G,G_1,Y) for any subset Y.

The core contribution is the definition of the Pesin–Pitskel topological pressure for NAIFS using the Carathéodory–Pesin structure. For a continuous potential φ∈C(X,ℝ), the authors consider (n,δ)-Bowen balls B_n(x,δ) built from the metric d_n (the analogue of d_max^n) and the Birkhoff sums S_w φ(x) over all words w of length n. They introduce covering sums M(Z,f,φ,α,δ,N) and R(Z,f,φ,α,δ,N) that weight each Bowen ball by exp(−α n + S_n φ(x)) (or its supremum over the ball). Taking limits as N→∞ yields functions m, r, and \bar r, whose critical values define respectively the Pesin–Pitskel pressure P_Z(f,φ) and the lower and upper capacity pressures C_P_Z(f,φ), \overline{C}_P_Z(f,φ). The authors prove that these definitions are robust: alternative formulations using the supremum of φ over Bowen balls lead to the same limits, and the pressures are monotone in δ, allowing the final limits δ→0.

A crucial theorem shows that the Pesin–Pitskel pressure coincides with the previously studied weighted topological pressure for NAIFS. The proof hinges on the uniform continuity of φ: for any δ, the difference between S_n φ(x) and its supremum over B_n(x,δ) is bounded by n ε(δ) with ε(δ)→0 as δ→0. This yields the inequalities P_Z(f,φ,δ) ≤ P’_Z(f,φ,δ) ≤ P_Z(f,φ,δ)+ε(δ), which collapse to equality in the limit. Consequently, the new pressure inherits all known properties of the weighted pressure, including variational characterizations in special cases.

The second major result is a variational principle for the Pesin–Pitskel pressure on arbitrary non‑empty compact subsets K⊂X. For a Borel probability measure μ supported on K, the measure‑theoretic pressure is defined as
 P_μ(f,φ) = h_μ(f) + ∫ φ dμ,
where h_μ(f) is the non‑autonomous analogue of Kolmogorov–Sinai entropy. The authors prove that
 P_K(f,φ) = sup{ P_μ(f,φ) : μ ∈ 𝔐(K) },
where 𝔐(K) denotes all Borel probability measures supported on K. The proof follows the classical strategy: the upper bound uses (n,ε)-separated sets to construct measures with large entropy, while the lower bound employs (n,ε)-spanning sets and the Carathéodory construction to approximate any covering sum by measures. The equicontinuity of the NAIFS and the continuity of φ guarantee that the approximations become exact as n→∞ and ε→0.

Overall, the paper establishes a full thermodynamic formalism for NAIFS that aligns the maximal (Biś) entropy perspective with pressure theory. By proving the equivalence with weighted pressure and delivering a clean variational principle, the authors bridge the gap between free semigroup actions and non‑autonomous dynamics, providing a unified framework applicable to multifractal analysis, dimension theory, and the study of random or time‑varying fractal constructions. The results open avenues for further exploration of equilibrium states, uniqueness of Gibbs measures, and multifractal spectra in the context of time‑dependent iterated function systems.