Almgren-Poon Parabolic Frequency Function

• The paper establishes a variational principle linking packing topological pressure with measure-theoretic local pressures in nonautonomous dynamical systems.
• It employs Carathéodory covering and Billingsley-type techniques to extend Bowen–Pesin–Pitskel’ theory to nonautonomous, packing-pressure contexts.
• The approach has practical implications for multifractal analysis, dimension theory, and random dynamics, enhancing our understanding of complex system behaviors.

The variational principle for packing topological pressure in nonautonomous dynamical systems (NDS) establishes a precise correspondence between a dynamically-defined topological invariant—the packing topological pressure—and a supremum of measure-theoretic local pressure functionals over all probability measures supported on a given compact set. This framework unifies and extends earlier pressure distribution principles, such as the Bowen–Pesin–Pitskel’ principle, to accommodate the inherent nonautonomy in sequences of continuous self-maps, and it facilitates applications in dimension theory, multifractal analysis, and random dynamics (Li, 2024).

1. Setting and Definitions

Consider a compact metric space $(X,d)$ and a sequence of continuous selfmaps $f_1, f_2, \dots$ on $X$, forming an NDS. For $n\in\mathbb N$, let $f_1^n = f_n \circ f_{n-1} \circ \cdots \circ f_1$, and $S_n\varphi(x) = \sum_{k=0}^{n-1} \varphi(f_1^k(x))$ for any potential $\varphi\in C(X)$.

Given a nonempty compact $K\subset X$ and $\varepsilon>0$, the $n$-step (nonautonomous) Bowen ball is

$$B_n(x,\varepsilon) = \{y \in X: d(f_1^k(x), f_1^k(y)) < \varepsilon \ \text{for} \ 0 \leq k < n\}.$$

Packing topological pressure $PP(f_{1,\infty}, K,\varphi)$ is defined using Carathéodory-type coverings by pairwise disjoint Bowen balls, controlled by a parameter $s$ with the critical value taken as the supreme $s$ for which a certain weighted sum diverges as the covering length diverges and the scale parameter $\varepsilon\to0$. The precise construction parallels the Feng–Huang (packing) pressure, incorporating pairwise disjointness for the involved Bowen balls.

2. Relation to Other Pressures and Capacity Types

The formalism distinguishes several pressure-like invariants on arbitrary $K\subset X$, each constructed via variants of the Carathéodory–Pesin structure:

• Classical (Bowen–Ruelle) topological pressure: uses spanning or separated sets,
• Carathéodory–Pesin pressure ($P^B$),
• Lower/upper capacity pressures,
• Packing topological pressure ($PP$) [Feng–Huang type].

These quantities satisfy inequalities:

• $P^B \leq CPP \leq CP^U$,
• $P^B \leq PP \leq P$,
• $PP \leq CPP$,
• $CP^U = P$.

Thus, packing topological pressure is strictly intermediate between the Pesin (Carathéodory) pressure and the full topological pressure, and is generally distinct in the nonautonomous context [(Li, 2024), Thm 3.4].

3. Measure-Theoretic Local Pressure Functionals

Let $\mu$ be a Borel probability measure on $X$ (not required invariant under any of the $f_n$). The (upper and lower) local measure-theoretic pressure at $x\in X$ with respect to $\varphi$ is defined as

$$\overline{P}_\mu(f_{1,\infty}, x, \varphi) = \lim_{\varepsilon\to0} \limsup_{n\to\infty} \frac1n \left[ -\log \mu(B_n(x,\varepsilon)) + S_n\varphi(x) \right].$$

The global (upper) measure-theoretic pressure of $\mu$ on $K$ is $$\overline{P}_\mu(f_{1,\infty}, K, \varphi) = \int_K \overline{P}_\mu(f_{1,\infty}, x, \varphi) d\mu(x)$$.

4. The Variational Principle for Packing Pressure

The main result (Theorem 4.13 in (Li, 2024)) states:

For any NDS $(X, f_{1,\infty})$, any continuous potential $\varphi\in C(X)$, and any nonempty compact set $K\subset X$ with $PP(f_{1,\infty}, K, \varphi) > \sup|\varphi|$, we have $$> PP(f_{1,\infty}, K, \varphi) = \sup\{ \overline{P}_\mu(f_{1,\infty}, K, \varphi) : \mu\in M(X), \mu(K) = 1\}. >$$ Here the supremum is taken over all Borel probability measures supported on $K$. An equivalent expression is available in terms of the “packing pressure” $P_\mu$ (see (Li, 2024) for the detailed definition).

This generalizes the variational principle in the autonomous case (Bowen–Pesin–Pitskel’ pressure) to the nonautonomous, packing-pressure context, where neither topological invariance nor any form of Birkhoff or Shannon–McMillan–Breiman theorem is available.

5. Structure of the Proof and Distribution Principles

The proof relies on:

• A pressure distribution principle (for $P^B$): sequences of measures satisfying specified weighted upper bounds provide lower bounds for Pesin pressure.
• A Billingsley-type theorem (for $PP$): upper and lower bounds for packing pressure in terms of local measure-theoretic pressures.
• A Carathéodory covering (and extraction of a Cantor subsystem) to construct a measure supported on $K$ that nearly attains the packing pressure at almost every point, thereby closing the variational upper bound.

The equality is constructed by establishing both lower and upper bounds for $PP$ as a supremum over the family of upper-measure pressures $\overline{P}_\mu$.

6. Corollaries and Special Cases

• Invariant Measure Case: When all $f_n$ preserve a common Borel probability measure $\mu$, the principle recovers a classical variational principle for pressures over invariant measures.
• Generic Points: For $G_\mu\subset X$, the set of generic points for $\mu$ (i.e., points whose empirical measures converge to $\mu$), it is shown that $PP(f_{1,\infty}, G_\mu, \varphi)$ admits an explicit upper bound in terms of finite-step topological pressures over tubes of generic points, with sharper expressions available in the ergodic case [(Li, 2024), Prop 5.5].

7. Position in the Theory and Broader Connections

This variational principle:

• Extends Bowen–Pesin–Pitskel’ variational principles to packing pressure in NDS, confirming the deep connection between topological complexity measures and measure-theoretic growth rates even in highly nonautonomous, time-dependent settings.
• Adopts both Carathéodory-Pesin covering methods and Billingsley-type local-pressure tools to deal with the breakdown of invariance and ergodic theorems.
• Enables the analysis of multifractal spectra, dimension theory, and complexity in random and nonautonomous systems, providing foundational tools for modern nonautonomous ergodic theory (Li, 2024).

Type of Pressure	Extremal Formula	Reference Principle
(Autonomous) Topological	$\sup_{\mu} [h_\mu + \int \varphi d\mu]$	Ruelle–Walters/Bowen–Pesin–Pitskel’
(Autonomous) Packing	$\sup_{\mu} \overline{P}_\mu$	Feng–Huang, multifractal formalism
NDS Packing	$\sup_{\mu} \overline{P}_\mu$	(Li, 2024), Billingsley-type

The variational principle for packing topological pressure in nonautonomous settings thus forms a bridge between fine-scale topological invariants and local measure-theoretic growth rates, incorporating probabilistic, geometric, and combinatorial elements necessary to analyze complex time-dependent dynamics.