Variational principles for nonautonomous dynamical systems

This paper establishes variational principles for pressure functions in discrete nonautonomous dynamical systems on compact metric spaces by applying methods from convex analysis to the thermodynamical formalism.

The Gist

Imagine you are trying to understand the behavior of a complex machine, like a giant, chaotic weather system or a bustling city. In mathematics, we call these "dynamical systems." Usually, scientists study systems that follow the same rules every day (autonomous systems). But what if the rules change every single day? Maybe the wind blows differently on Monday than on Tuesday, or the city's traffic lights change their timing every hour. This is a Nonautonomous Dynamical System (NADDS).

This paper by Andrzej Biś is a mathematical toolkit designed to measure the "chaos" and "energy" of these ever-changing systems. Here is the breakdown using simple analogies.

1. The Problem: The Moving Target

In the old days, mathematicians had a perfect ruler to measure chaos, called Topological Entropy. Think of it like a speedometer for a car that drives on a straight, unchanging road. It tells you how fast the car is going.

But in a Nonautonomous system, the road keeps changing shape, the gravity shifts, and the speed limits vary every second. The old speedometer breaks. Also, usually, we look for a "steady state" (a measure that stays the same over time). In these changing systems, a steady state might not even exist. It's like trying to find a calm spot in a hurricane that keeps changing direction.

2. The Solution: The "Pressure" Gauge

The author introduces a new concept called Topological Pressure.

• The Analogy: Imagine you are a chef trying to predict how a soup will taste.
  • Entropy is just the temperature of the soup (how chaotic it is).
  • Pressure is the temperature plus the flavor of the ingredients (the "potential").
  • If you add a spicy ingredient (a high-value potential), the "pressure" of the soup changes.

The paper proves that even if the recipe changes every minute (the nonautonomous part), we can still calculate this "Pressure" and find a mathematical relationship between the chaos of the system and the ingredients we add.

3. The Core Discovery: The Variational Principle

The most famous part of the paper is the Variational Principle. In the world of classical physics, there is a rule that says: "Nature always chooses the path of least resistance."

In this math paper, the author proves a similar rule for these chaotic, changing systems:

The total "Pressure" of the system is equal to the maximum possible sum of "Chaos" (Entropy) and "Flavor" (Potential) across all possible ways the system could behave.

The Metaphor:
Imagine a massive ballroom dance floor where the music changes every 10 seconds.

• The Dancers (Measures): There are many ways the dancers could move. Some move wildly (high chaos), some move in a slow, steady rhythm (low chaos).
• The Music (Potential): The music might be fast or slow, loud or quiet.
• The Goal: The system naturally "wants" to find the specific dance style that maximizes the fun (Pressure).

The paper proves that no matter how crazy the music changes, there is always a "best dance" (a specific mathematical measure) that balances the wildness of the moves with the rhythm of the music to create the maximum possible energy.

4. Two Types of Pressure

The author doesn't just stop at one definition. He creates two different ways to measure this, like using two different types of thermometers:

1. Topological Pressure: This is the standard way, looking at how many different paths the system can take.
2. Misiurewicz Pressure: This is a more specialized, "fine-tuned" version. It looks at the system through a very specific lens (using neighborhoods and uniform structures), similar to how a microscope reveals details a regular eye misses.

The paper proves that both of these thermometers work perfectly. They both follow the same "Variational Principle" rule: Maximize the sum of chaos and flavor.

5. The "Ghost" of Invariance

One of the hardest parts of studying these systems is that they often don't have a "steady state" (an invariant measure). It's like trying to find a shadow that never moves, even though the sun is spinning wildly.

The author shows that even if a perfect steady state doesn't exist, we can still define a "virtual" steady state. He proves that if you find the dance style that maximizes the Pressure, that specific dance style automatically becomes a "steady" pattern for the system, even if the rules keep changing. It's as if the dancers, by trying to maximize their fun, accidentally synchronize themselves into a perfect, invariant rhythm.

Summary

In plain English, this paper says:
"Even if a system changes its rules every single second and has no stable pattern, we can still measure its complexity. We can prove that there is always a 'best' way to describe the system's behavior that balances its randomness with the forces acting on it. We can do this using two different mathematical methods, and both methods lead to the same beautiful, predictable conclusion."

It's a triumph of order over chaos, showing that even in a world of constant change, there are deep, unbreakable laws governing how things evolve.

Technical Summary

Here is a detailed technical summary of the paper "Variational Principles for Nonautonomous Dynamical Systems" by Andrzej Bi´s.

1. Problem Statement and Context

The paper addresses the extension of thermodynamical formalism from classical autonomous dynamical systems to nonautonomous discrete dynamical systems (NADDS).

• Classical Context: In classical dynamics (a single map $f: X \to X$), the Variational Principle relates topological pressure $P_{top}(f, \phi)$ to measure-theoretic entropy $h_\mu(f)$ via the formula:
  $$P_{top}(f, \phi) = \sup_{\mu \in \mathcal{P}_f(X)} \left\{ h_\mu(f) + \int_X \phi \, d\mu \right\}$$
  where $\mathcal{P}_f(X)$ is the space of $f$-invariant probability measures.
• The Challenge in NADDS: A NADDS is defined by a sequence of continuous self-maps $f_{1,\infty} = \{f_n: X \to X\}_{n \in \mathbb{N}}$ on a compact metric space $X$. Unlike the autonomous case, NADDS generally lack a common invariant measure (i.e., a measure $\mu$ such that $f_n \mu = \mu$ for all $n$). The classical Krylov-Bogoliubov theorem does not apply, making the standard definition of invariant measures and the direct application of variational principles problematic.
• Goal: To establish rigorous variational principles and dual variational principles for the topological pressure and a newly defined "Misiurewicz pressure" in the nonautonomous setting, without relying on the existence of common invariant measures.

2. Methodology

The author employs a Convex Analysis approach, utilizing abstract results from a previous work by Bi´s et al. [2]. The methodology proceeds as follows:

1. Abstract Pressure Functions: The paper defines a "pressure function" $\Gamma: \mathcal{B} \to \mathbb{R}$ on a Banach space $\mathcal{B}$ (typically $C(X)$ or $L^\infty(X)$) satisfying three axioms:
   • Monotonicity: $\phi \leq \psi \implies \Gamma(\phi) \leq \Gamma(\psi)$.
   • Translation Invariance: $\Gamma(\phi + c) = \Gamma(\phi) + c$.
   • Convexity: $\Gamma(t\phi + (1-t)\psi) \leq t\Gamma(\phi) + (1-t)\Gamma(\psi)$.
2. Abstract Variational Principle (Theorem 3.3): Using the properties of convex functions and the Riesz representation theorem, the paper establishes that for any such pressure function, there exists an upper semi-continuous entropy map $h: \mathcal{P}(X) \to \mathbb{R}$ (where $\mathcal{P}(X)$ is the space of Borel probability measures) such that:
   $$\Gamma(\phi) = \max_{\mu \in \mathcal{P}(X)} \left\{ h(\mu) + \int_X \phi \, d\mu \right\}$$
   Crucially, this result holds without assuming the existence of invariant measures; it applies to the space of all probability measures.
3. Application to NADDS: The author verifies that both the Topological Pressure and the Misiurewicz Pressure defined for NADDS satisfy the axioms of an abstract pressure function. This allows the direct application of the abstract variational principle to these specific dynamical contexts.

3. Key Definitions

• Topological Pressure ($P_{top}$): Defined via separated sets and the sequence of metrics $d_n$ generated by the composition of maps $f_1, \dots, f_{n-1}$.
• Misiurewicz Pressure ($P_{Mis}$): Inspired by Misiurewicz's work on $\mathbb{Z}^n_+$ actions, this is defined using neighborhoods of the diagonal in $X \times X$ (uniform structure) and $(n, \delta)$-separated/spanning sets. It serves as an alternative pressure definition.
• Abstract Measure Entropy ($h_{f_{1,\infty}}$): A function derived from the pressure function via the dual formula:
  $$h_{f_{1,\infty}}(\mu) = \inf_{\phi \in C(X)} \left\{ P_{top}(f_{1,\infty}, \phi) - \int_X \phi \, d\mu \right\}$$
• Invariant Measures ($\mathcal{P}_{f_{1,\infty}}(X)$): The set of measures $\mu$ such that $\int (\psi \circ f_n) d\mu = \int \psi d\mu$ for all $n$. The paper notes this set may be empty.

4. Key Results and Theorems

A. Abstract Variational Principles (Theorems A & C)

The paper proves that for both Topological Pressure and Misiurewicz Pressure, there exists a unique upper semi-continuous entropy map $h$ such that the pressure is the maximum of the sum of entropy and the integral of the potential over all Borel probability measures (not just invariant ones).

• Theorem A (Topological Pressure):
  $$P_{top}(f_{1,\infty}, \phi) = \max_{\mu \in \mathcal{P}(X)} \left\{ h_{f_{1,\infty}}(\mu) + \int_X \phi \, d\mu \right\}$$
• Theorem C (Misiurewicz Pressure):
  $$P_{Mis}(f_{1,\infty}, \phi) = \max_{\mu \in \mathcal{P}(X)} \left\{ h_{Mis}(\mu) + \int_X \phi \, d\mu \right\}$$
• Uniqueness (Corollaries 1 & 2): If the pressure function is Gâteaux differentiable at a potential $\phi$, the maximizing measure $\mu$ is unique.

B. Relation to Invariant Measures (Theorems B & D)

The paper investigates the relationship between the abstract entropy $h$ and the set of invariant measures $\mathcal{P}_{f_{1,\infty}}(X)$.

• Characterization of Invariance: A measure $\mu$ is invariant ($\mu \in \mathcal{P}_{f_{1,\infty}}(X)$) if and only if its abstract entropy is non-negative ($h(\mu) \geq 0$).
• Maximization over Invariant Measures: If the set of invariant measures is non-empty and a "classical" variational principle holds over this subset, then the maximum over all measures coincides with the maximum over invariant measures.
  $$P_{top}(f_{1,\infty}, \phi) = \max_{\mu \in \mathcal{P}_{f_{1,\infty}}(X)} \left\{ h_{f_{1,\infty}}(\mu) + \int_X \phi \, d\mu \right\}$$
• Entropy Inequalities: The paper establishes a hierarchy of entropies for invariant measures: $0 \leq h(\mu) \leq h^(\mu) \leq h_{f_{1,\infty}}(\mu)$, where$h^$ is a star-entropy defined via limits of invariant measures.

5. Significance and Contributions

1. Overcoming the Lack of Invariant Measures: The primary contribution is providing a rigorous variational framework for NADDS even when no common invariant measure exists. By shifting the maximization domain to the space of all probability measures $\mathcal{P}(X)$, the theory remains valid.
2. Unified Approach via Convex Analysis: The paper demonstrates the power of abstract convex analysis in dynamical systems. By isolating the properties of the pressure function (monotonicity, translation invariance, convexity), the author derives deep dynamical results without needing specific dynamical constructions for every new system type.
3. Dual Variational Principles: The paper establishes the dual relationship (recovering entropy from pressure), which is essential for the study of equilibrium states and Gibbs measures in nonautonomous settings.
4. Misiurewicz Pressure: The introduction and analysis of Misiurewicz pressure for NADDS provides a new tool for studying these systems, offering an alternative to the standard topological pressure definition, particularly useful for systems where uniform structures play a key role.
5. Characterization of Equilibrium States: The results clarify that "equilibrium states" (measures maximizing the variational expression) are necessarily invariant measures if the entropy is non-negative, bridging the gap between abstract optimization and physical invariance.

In summary, the paper successfully generalizes the thermodynamic formalism to nonautonomous systems, proving that the fundamental link between pressure and entropy holds universally, provided one utilizes the correct abstract definitions and the full space of probability measures.