---
title: Almgren-Poon Parabolic Frequency Function
url: https://www.emergentmind.com/topics/parabolic-frequency-function-of-almgren-poon-type
type: topic
---

# Almgren-Poon Parabolic Frequency Function

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 [2412.17634].

## 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 [2412.17634, 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 [2412.17634]) 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 [2412.17634] 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 [2412.17634, 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 [2412.17634].

| 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             | $\sup_{\mu} \overline{P}_\mu$ | [2412.17634], 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.

Source: https://www.emergentmind.com/topics/parabolic-frequency-function-of-almgren-poon-type