Induced and nonlinear topological pressure for random dynamical systems

Abstract: In this paper, we investigate induced and nonlinear fiber topological pressure for random dynamical systems. We define a non-averaged induced fiber pressure via spanning and separated sets, characterize it as the pseudo-inverse of the classical fiber topological pressure studied previously, and establish the corresponding variational principle. We also define the nonlinear fiber pressure and prove the associated variational principles. Finally, we extend the combined theory to the higher-dimensional setting.

• The paper introduces a novel framework establishing variational principles for induced and nonlinear fiber topological pressure in random dynamical systems.
• It extends classical thermodynamic formalism with a pseudo-inverse characterization and critical exponent formulation applied to stochastic processes.
• It proves continuity, strict monotonicity, and convexity properties that facilitate numerical methods for modeling multifractal spectra and phase transitions.

Induced and Nonlinear Topological Pressure for Random Dynamical Systems

Overview and Context

This paper (2604.14964) systematically develops the theory of induced and nonlinear fiber topological pressure in the context of random dynamical systems, extending the classical thermodynamic formalism. It provides new definitions, critical exponent formulations, and variational principles for both induced and nonlinear forms of topological pressure, ensuring their applicability to systems governed by stochastic processes on fiber bundles. The work draws together techniques from topological dynamics, ergodic theory, and statistical physics, generalizing several deterministic results to random settings and higher-dimensional cases.

Classical and Induced Topological Pressure

The paper situates the classical notions of entropy ($h_\mu$) and topological pressure ($P(f, \varphi)$) within the framework of random dynamical systems. In classical deterministic systems, topological pressure unifies the concepts of measure-theoretic entropy and potential functions via the variational principle: $$P(f, \varphi) = \sup_{\mu \in M(X,f)} \left[ h_\mu(f) + \int_X \varphi\,d\mu \right]$$ Pressure can be formulated via open covers, spanning sets, or separated sets, and is intimately connected to equilibrium states which maximize the variational expression.

Induced pressure arises naturally when time is parametrized by a positive observable (scaling function $\psi$), and the pressure is taken over the induced-time partition. The induced pressure has its own variational principle: $$P_\psi(f, \varphi) = \sup_{\mu \in M(X,f)} \frac{h_\mu(f) + \int_X \varphi\,d\mu}{\int_X \psi\,d\mu}$$ The paper recalls that the induced pressure can be identified as the unique zero of the function $\beta \mapsto P(f, \varphi - \beta \psi)$, leading to a pseudo-inverse characterization of induced pressure.

Random Dynamical Systems and Fiber Pressure

Random dynamical systems consist of bundles of continuous maps $f_\omega:X \to X$ indexed by a base probability space $(\Omega, \mathcal{F}, P,\theta)$. Fiber topological pressure generalizes deterministic pressure via integration over the base: $$P_\Omega(f, \varphi) = \lim_{\epsilon \to 0} \limsup_{n \to \infty} \frac{1}{n} \int_\Omega \log Q_\omega(f, \varphi, n, \epsilon)\,dP(\omega)$$ where $Q_\omega$ denotes the partition function over spanning sets in fiber $P(f, \varphi)$0 for each $P(f, \varphi)$1. A variational principle analogous to the deterministic case holds: $P(f, \varphi)$2 This identifies the fiber pressure as the supremum of the fiberwise entropy and potential integral over all $P(f, \varphi)$3-invariant measures with fixed base marginal.

Induced Fiber Pressure: Definition and Properties

The paper introduces a non-averaged, measurable definition of induced fiber pressure via spanning and separated sets over induced-time partitions. The main variational principle is established: $P(f, \varphi)$4 A critical exponent formula is provided, characterizing induced pressure as the pseudo-inverse of classical fiber pressure: $P(f, \varphi)$5 The properties of induced pressure, including convexity of the set of equilibrium states and extremality of ergodic measures, extend classical notions to the random dynamical system framework.

Nonlinear Fiber Pressure and Variational Principle

The paper generalizes thermodynamic formalism to "nonlinear" pressure, where the usual Birkhoff sum in the exponent is replaced by a nonlinear function $P(f, \varphi)$6: $P(f, \varphi)$7 The corresponding variational principle holds under either convexity of $P(f, \varphi)$8 or abundance of ergodic measures: $P(f, \varphi)$9 This extends recent developments in nonlinear thermodynamic formalism, allowing for pressure functions that capture non-additive or multi-scale phenomena.

Higher-Dimensional Extensions

The paper further extends the theory to higher-dimensional potentials $$P(f, \varphi) = \sup_{\mu \in M(X,f)} \left[ h_\mu(f) + \int_X \varphi\,d\mu \right]$$0 and corresponding nonlinear functions $$P(f, \varphi) = \sup_{\mu \in M(X,f)} \left[ h_\mu(f) + \int_X \varphi\,d\mu \right]$$1. The variational principle for higher-dimensional nonlinear induced fiber pressure is proven: $$P(f, \varphi) = \sup_{\mu \in M(X,f)} \left[ h_\mu(f) + \int_X \varphi\,d\mu \right]$$2 The critical exponent characterization and equilibrium state structure carry over to this setting.

Numerical and Structural Results

The paper delivers strong formal results:

• The induced fiber pressure is strictly decreasing in the parameter $$P(f, \varphi) = \sup_{\mu \in M(X,f)} \left[ h_\mu(f) + \int_X \varphi\,d\mu \right]$$3, ensuring the existence and uniqueness of zeros.
• Continuity of pressure functions with respect to $$P(f, \varphi) = \sup_{\mu \in M(X,f)} \left[ h_\mu(f) + \int_X \varphi\,d\mu \right]$$4 and supremum norms is rigorously proven.
• The convex structure and ergodic decomposition of equilibrium states are preserved.
• The critical exponent formulation enables direct computation of induced (nonlinear and fiber) pressures via standard pressure functions.

Implications and Future Directions

These results broaden the scope of thermodynamic formalism to settings where randomness and nonlinearity play critical roles. Applications span statistical mechanics, complex systems, ergodic optimization, and multifractal analysis, especially in circumstances involving random environments or noise. The extension to higher-dimensional and nonlinear potentials enables modeling of interacting observables and non-standard scaling, which may facilitate analysis of multifractal spectra and phase transitions in random systems. The analogy between induced pressure and critical points of pressure functions suggests powerful numerical approaches for computation and inference.

Future extensions may include:

• Fine-grained equilibrium state theory for specific classes of random dynamical systems, such as those with countable Markov shifts or non-uniform hyperbolicity.
• Application to stochastic models in statistical physics or information theory, addressing multifractal decomposition under randomness and nonlinear scaling.
• Investigation of phase transitions and universality classes in systems with higher-dimensional potentials and random forcing.
• Algorithmic implications for estimating induced and nonlinear fiber pressure in large-scale random systems.

Conclusion

The paper presents a comprehensive formalism for induced and nonlinear topological pressure in random dynamical systems, establishing variational principles, critical exponent characterizations, and higher-dimensional extensions. These results firmly anchor the extension of classical thermodynamic formalism to stochastic and non-additive dynamical settings, providing a robust mathematical framework with both theoretical and practical significance.