SRB Entropy Functional in Dynamics

• SRB entropy functional is a measure quantifying the balance between metric entropy and local expansion rates in smooth dynamical systems.
• It characterizes equilibrium states by identifying invariant measures that saturate Ruelle’s inequality via Lyapunov exponents and Jacobian functions.
• Its applications extend to non-uniformly expanding maps, partial hyperbolicity, and even infinite-dimensional Banach space systems.

The SRB entropy functional is a measure-theoretic and thermodynamic object that plays a central role in the modern theory of smooth dynamical systems, particularly in the classification, characterization, and statistical description of invariant measures with physical (observable) relevance. Originating from the paper of Sinai–Ruelle–Bowen (SRB) measures and their associated entropy, the SRB entropy functional formalizes the admissible balance between metric entropy and local expansion rates, typically encoded via Lyapunov exponents or Jacobian-like functions. Its extremal properties characterize natural invariant measures (SRB or SRB-like) as equilibrium states, unify entropy production with volume growth, and provide a variational principle for nonequilibrium steady states, especially in non-invertible, non-uniformly expanding, or more general settings.

1. Fundamental Definition and Thermodynamic Variational Principle

The SRB entropy functional is defined on the space of invariant probability measures of a smooth map, most classically for $C^{1+\alpha}$ diffeomorphisms or expanding maps, but extended to $C^1$ and even to random and infinite-dimensional (Banach space) dynamical systems. For a dynamical system $f:M\to M$, given an $f$-invariant Borel probability measure $\mu$, the functional is expressed as

$F(\mu) = h_\mu(f) - \int \varphi\, d\mu,$

where $h_\mu(f)$ is the metric entropy of $f$ with respect to $\mu$, and $\varphi$ is a dynamically relevant potential — typically $\varphi = \log |f'|$ (or the logarithm of the unstable Jacobian) in the expanding or hyperbolic setting.

For $C^1$ expanding maps $f:S^1\to S^1$ (i.e., $|f'(x)|>1$ everywhere), the entropy functional specializes to

$F(\mu) = h_\mu(f) - \int \log|f'|\, d\mu,$

and relates to the pressure $P_f$ of the potential $-\log|f'|$ via the variational principle: $$P_f = \sup_{\mu\in\mathcal{M}_f} \left[ h_\mu(f) + \int -\log|f'|\, d\mu \right] = 0,$$ so the maximizers—i.e., those with $F(\mu)=0$—are precisely the equilibrium states for $-\log|f'|$. This observation extends to more general contexts, such as non-uniformly expanding maps and systems with inducing schemes.

2. Characterization of SRB and SRB-like Measures via the Entropy Functional

Statistically significant invariant measures are those for which the SRB entropy functional achieves its maximum (i.e., $F(\mu)=0$). These are exactly the equilibrium states for the given potential. In classical hyperbolic theory:

• SRB measures are invariant measures whose empirical basins have positive Lebesgue measure, and whose conditional measures on unstable manifolds are absolutely continuous with respect to induced Riemannian volume.
• SRB-like measures (a generalization necessary in the low regularity $C^1$ context) are those whose statistical basins (ε-basins of weak attraction) for the sequence of empirical measures have positive Lebesgue measure for all $\epsilon>0$. In general, SRB-like measures exist always, and when SRB measures exist they coincide.

Crucially, for SRB and SRB-like measures,

$$h_\mu(f) = \int \log|f'|\, d\mu, \quad \text{or more generally,} \quad h_\mu(f) = \int \sum_{\lambda_i > 0} m_i(x)\, \lambda_i(x) d\mu,$$

(a form of Pesin's entropy formula). The entropy functional thus selects those measures that strictly saturate Ruelle's inequality. The set of such measures is weak* compact and convex, with ergodic (extremal) points corresponding to ergodic SRB-like measures (Catsigeras et al., 2012, Araujo et al., 2017).

3. Generalizations and Extensions in Regularity and Geometric Setting

The SRB entropy functional and its variational characterization extend in several important directions:

• Non-uniform expansion and partial hyperbolicity: For $C^1$ non-uniformly expanding maps with hyperbolic times, and for partially hyperbolic attractors with non-uniform expansion in cone fields, SRB-like and SRB measures are shown to exist and to satisfy Pesin's formula, often via inducing schemes or Young towers (Araujo et al., 2017, Cruz et al., 2018, Alves et al., 2021).
• Banach and infinite-dimensional settings: For $C^2$ mappings of Banach spaces or random Banach dynamical systems, the entropy functional is equal to the volume growth on unstable manifolds, i.e., the sum of positive Lyapunov exponents:

$$h_\mu(f) = \int \sum_{i} m_i(x) \lambda_i^+(x) \, d\mu,$$

provided technical conditions—like finite box-counting dimension of the invariant set—are imposed to ensure partition properties needed for the definition and comparison of entropy (Blumenthal et al., 2015, Luo et al., 2022).

• Non-invertible and inverse entropy theory: For endomorphisms, the forward SRB entropy functional captures the sum of positive exponents, while the inverse entropy associates to the backwards (contracting) directions, leading to rigidity phenomena where forward and backward entropy classify the system up to conjugacy (Mihailescu et al., 9 Mar 2025).

4. Structural and Rigidity Properties

A series of rigidity results demonstrate that the SRB entropy functional encodes significant structural information about the system:

• In DA-diffeomorphisms isotopic to Anosov maps on tori, the unique measure of maximal entropy is SRB if and only if the sum of positive Lyapunov exponents matches that of the linear model on all periodic orbits in the support (Micena et al., 8 Apr 2024).
• Entropy rigidity implies smooth conjugacy between systems under the equivalence of both forward and inverse SRB entropy values (Mihailescu et al., 9 Mar 2025).
• In families of systems, the SRB entropy functional (and thus the entropy of the SRB measure) varies continuously under smooth perturbations (statistical stability), provided uniform hyperbolicity and dominated splitting are maintained (Cruz et al., 2018).

This positions the SRB entropy functional as both a classifier and invariant for families of dynamical systems—its constancy, vanishing, or extremal status identifies rigidity or smooth equivalence classes.

5. Differentiability, Gradient Flows, and Linear Response

The SRB entropy functional is differentiable when the underlying family of maps is modeled as a Hilbert manifold with sufficient Sobolev regularity (specifically, $k>3+n/2$ for maps on the $n$-torus), by virtue of the Sobolev embedding theorem ensuring $C^3$ regularity.

• Its Fréchet derivative has an explicit formula involving the derivative of the Jacobian and the linear response (Fréchet derivative) of the SRB measure with respect to perturbations in the map. For expanding maps on the torus:

$$D(f)g = \int_{T^n} (Df)^{-1} Dg \cdot \rho \, dx + \sum_{n=0}^\infty \int_{T^n} \mathcal{L}_f^n[g\rho](x) \cdot \log Jf(f^n x) \, dx,$$

where $\mathcal{L}_f$ is the transfer operator and $\rho$ the SRB density.

• The existence of a Lipschitz continuous gradient vector field allows formulation and local integration of a gradient flow for the SRB entropy, yielding a (short-time) variational evolution toward maximizing (or minimizing) SRB entropy (Chen et al., 23 Sep 2025).
• The link between the derivative of the SRB entropy and the linear response of the SRB measure forms a bridge between ergodic theory and thermodynamic/variational methods and underpins sensitivity analysis of statistical properties under perturbation.

6. Inducing Schemes, Weakening of Ruelle's Inequality, and Pathology

When classical (full regularity) results fail, the SRB entropy functional retains significance:

• SRB measures constructed by inducing schemes (Gibbs–Markov maps, Young towers) in piecewise smooth or singular regimes still obey Pesin-type entropy formulas, under suitable integrability of return times and local expansion, but the Ruelle inequality can fail or even reverse if entropy control is lost (Alves et al., 2021).
• Examples demonstrate systems where the metric (Kolmogorov–Sinai) entropy is infinite (despite finite average Jacobian), thus violating the expectations from Ruelle's inequality and showing the domain of the entropy functional is strictly limited by partition and integrability structure.

7. Formal Group Perspective and Universality

Abstracting entropy as a functional on probability spaces subject to composability and extensivity axioms, group-theoretic frameworks (e.g., group entropies) recover the SRB entropy functional as the distinguished solution for systems with exponential partition growth, i.e., when the multiplicity of accessible states is exponential in time or system size (Jensen et al., 9 Jul 2025): $S[p] = \sum_{i} p_i \log \frac{1}{p_i}$ up to scaling, matches the SRB (or Kolmogorov–Sinai) entropy rate for chaotic dynamical systems. The framework suggests how to generalize the entropy functional in systems with non-exponential growth or complex composability, offering a unifying conceptual underpinning.

---

Summary Table: Forms of the SRB Entropy Functional in Various Settings

Setting	SRB Entropy Functional Formula	Validity/Comments
$C^1$ expanding ($f:S^1\to S^1$)	$F(\mu) = h_\mu(f) - \int \log|f'|\,d\mu$	Maximized at SRB/SRB-like measures
$C^1$ non-uniform expansion	$h_\mu(f) + \int -\log|\det Df|\,d\mu = 0$	Holds for weak-SRB-like measures; generalizes Pesin
Banach/Random dynamical systems	$$h(\mu|P) = \int \sum_i m_i(\omega,x) \lambda_i^+ d\mu$$	Sum over positive Lyapunov exponents
Partial hyperbolicity (invertible)	$$h_\mu(f) = \int \log \frac{|\det Df|}{|\det Df|_{E^s}|} d\mu$$	Stable and center-unstable splits
Noninvertible/Gibbs–Markov (SRB measure)	$h_\mu(f) = \int \log|\det Df|\, d\mu$	Under induced schemes with integrability
Inverse SRB measure (repellor)	$$h_\text{inv}(\mu^-)= -\sum_{i: \lambda_i^-<0} \lambda_i^-$$	Entropy for backwards/contracting dynamics

---

The SRB entropy functional thus acts as a universal selector of physically and statistically meaningful invariant measures, governs the relation between dynamical complexity and local instability, and provides a foundational tool in both the ergodic and thermodynamic analysis of chaotic dynamical systems. Rigorous extension to degenerate regularity, geometric generality, or non-invertibility crucially depends on ensuring the validity of partition theorems, integrability of inducing schemes, or other structural controls which support the meaningful application and interpretation of the SRB entropy functional.