Pesin's Theorem: Entropy & Lyapunov Exponents

• Pesin's Theorem is a fundamental result that equates the metric (Kolmogorov–Sinai) entropy of an invariant measure with the space-average sum of positive Lyapunov exponents in differentiable dynamical systems.
• The theorem extends to C¹ settings using dominated splitting and underpins the existence of SRB-like measures in systems where regularity conditions are relaxed.
• Applications of Pesin’s Theorem span continuous-time flows, random dynamical systems, and infinite-dimensional models, significantly advancing modern ergodic theory and thermodynamic formalism.

Pesin’s Theorem is a fundamental result in smooth ergodic theory establishing a precise relationship between the metric entropy of an invariant measure and the sum of positive Lyapunov exponents for a broad class of differentiable dynamical systems. Formulated originally in the context of $C^{1+\alpha}$ or $C^2$ diffeomorphisms with hyperbolic invariant measures, Pesin's formula states that the Kolmogorov–Sinai entropy of a measure equals the space-average of the sum of positive Lyapunov exponents—counted with multiplicity—of the system. The theorem underpins much of the modern thermodynamic formalism and has been generalized to various settings including $C^1$ systems with dominated splitting, non-uniformly expanding maps, random dynamical systems, flows, and even infinite-dimensional spaces.

1. Classical Formulation and the Entropy–Exponent Link

For a $C^{1+\alpha}$ or $C^2$ diffeomorphism $f: M \to M$ of a compact Riemannian manifold, and an $f$-invariant ergodic probability measure $\mu$, Oseledets’ theorem yields, for $\mu$-a.e. $x$, a decomposition of the tangent space $C^2$0 into $C^2$1-invariant subspaces with Lyapunov exponents $C^2$2. The metric entropy $C^2$3 (Kolmogorov–Sinai entropy) measures the average exponential complexity of the orbit structure with respect to $C^2$4. Pesin’s formula asserts:

$C^2$5

This equality, proved by Pesin (1977), connects the information-theoretic entropy to the sum of positive Lyapunov exponents and is typically realized by physical or Sinai–Ruelle–Bowen (SRB) measures (Araujo et al., 2017).

2. Extension to $C^2$6 Dynamics and Dominated Splitting

The extension of Pesin’s theorem to $C^2$7 diffeomorphisms requires substituting regularity assumptions with geometric ones, namely dominated splitting. A $C^2$8-invariant splitting $C^2$9 is dominated if there exist $C^1$0, $C^1$1 such that for all $C^1$2 and $C^1$3,

$C^1$4

where $C^1$5 is the co-norm. Provided the absence of "mixed behavior" (no positive Lyapunov exponents on $C^1$6 and no negative on $C^1$7), and focusing on topological attractors with this splitting, Yang and Cao establish the existence of an invariant measure $C^1$8 satisfying Pesin’s entropy formula in the $C^1$9 setting (Yang et al., 2015):

$C^{1+\alpha}$0

The proof employs the variational principle for the potential $C^{1+\alpha}$1, upper semicontinuity of entropy, and uniform control of distortion on $C^{1+\alpha}$2-plaques via domination.

Similar lower-bound results are obtained in (Sun et al., 2010) and (Catsigeras et al., 2012) for $C^{1+\alpha}$3 diffeomorphisms with absolutely continuous invariant measures or SRB-like measures and dominated splitting. The equality is achieved when exponents along $C^{1+\alpha}$4 are nonnegative and along $C^{1+\alpha}$5 are nonpositive, enforcing the sign condition necessary for no mixed behavior.

3. Conditional Entropy Formulas for Unstable Foliations

For $C^{1+\alpha}$6 diffeomorphisms with dominated splitting, it is natural to consider a hierarchy of unstable foliations $C^{1+\alpha}$7 tangent to bundles corresponding to positive Lyapunov exponents. The entropy along such a foliation is defined via conditional measures and rates of contraction on plaques. Wang–Wang–Zhu (Wang et al., 2017) establish Pesin-type formulas for the conditional entropy $C^{1+\alpha}$8 along each foliation:

$C^{1+\alpha}$9

under the assumption that the conditional measures along $C^2$0 are absolutely continuous with respect to the Riemannian volume. This hierarchy gives precise correspondence with the global formula when the sum is taken over all positive exponents.

4. SRB-like Measures, Physical-like Invariant Measures, and Genericity

Pesin’s formula is commonly realized for SRB or SRB-like measures—measures whose statistical basins capture Lebesgue-typical orbits. In the $C^2$1 setting, a strict SRB measure may not exist, but SRB-like or physical-like invariant measures always exist for $C^2$2 expanding maps of the circle or Anosov diffeomorphisms (Catsigeras et al., 2012, Catsigeras et al., 2016). It is shown that:

• Every SRB-like measure satisfies Pesin’s entropy formula.
• The set of invariant measures satisfying the formula is the weak$C^2$3-closed convex hull of the ergodic SRB-like measures (Catsigeras et al., 2016).
• Generic $C^2$4 expanding maps possess a unique SRB-like measure which is not necessarily absolutely continuous, yet still satisfies the entropy formula (Catsigeras et al., 2012).

For $C^2$5 non-uniformly expanding maps, all ergodic weak-SRB-like measures are equilibrium states for the potential $C^2$6 and satisfy the entropy formula (Araujo et al., 2017).

5. Random Dynamical Systems and Infinite-Dimensional Extensions

Pesin’s formula generalizes to random and infinite-dimensional settings. For random $C^2$7 cocycles or random $C^2$8 dynamical systems in Banach spaces, under suitable integrability, one defines entropy as a conditional (fiber-wise) entropy and obtains

$C^2$9

where the exponents are those associated to the random Oseledets theorem (Biskamp, 2012, Luo et al., 2022).

In the context of finitely generated commuting $f: M \to M$0 diffeomorphisms on a separable Hilbert space, Li–Zhu (Li et al., 2020) extend the theory by introducing a higher-rank Oseledets theorem and recovering Pesin’s formula for SRB measures of random actions:

$f: M \to M$1

The structure of these measures and existence of local unstable manifolds in infinite dimensions depend on control of noncompactness and regularity assumptions.

6. Continuous-Time Flows and Hamiltonian Systems

The flow version of Pesin’s theorem for volume-preserving $f: M \to M$2 vector fields on three-manifolds, and Hamiltonian flows on four-manifolds, asserts that—generically in the $f: M \to M$3 topology—metric entropy of the flow equals the integrated sum of positive Lyapunov exponents:

$f: M \to M$4

for all $f: M \to M$5 (Bessa et al., 2010). The proof leverages continuity properties of entropy and Lyapunov exponents, and symbolic dynamics for flows in the Anosov and non-Anosov cases.

7. Significance, Limitations, and Applications

Pesin’s theorem provides a bridge between statistical complexity (entropy) and local instability (Lyapunov exponents), underpinning the measure-theoretic characterization of SRB-like measures, variational principles, and physicality in deterministic and random systems. The necessity of regularity is circumvented by geometric hypotheses such as dominated splitting, enabling the theorem to apply broadly to $f: M \to M$6 diffeomorphisms, dominated attractors, partially hyperbolic systems, and generic volume-preserving flows (Yang et al., 2015, Sun et al., 2010, Catsigeras et al., 2012, Bessa et al., 2010). However, absence of domination or control of the geometry of unstable leaves leads to failure of the formula in general $f: M \to M$7 dynamical systems, which marks a clear boundary for the theory.

Applications encompass symbolic extensions, asymptotic entropy expansiveness, and realization of Shub’s entropy conjecture for $f: M \to M$8 systems with dominated splitting (Yang et al., 2015). Random and infinite-dimensional generalizations further tie the formula to thermodynamic formalism in stochastic dynamics and high- or infinite-dimensional dynamical models (Luo et al., 2022, Li et al., 2020).

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Key references:

• "On Pesin's entropy formula for dominated splittings without mixed behavior" (Yang et al., 2015)
• "Formula of Entropy along Unstable Foliations for $f: M \to M$9 Diffeomorphisms with Dominated Splitting" (Wang et al., 2017)
• "Dominated Splitting and Pesin's Entropy Formula" (Sun et al., 2010)
• "Pesin Entropy Formula for C1 Diffeomorphisms with Dominated Splitting" (Catsigeras et al., 2012)
• "Pesin's Entropy Formula for $f$0 non-uniformly expanding maps" (Araujo et al., 2017)
• "Characterization of SRB Measures for Random Dynamical Systems in a Banach space" (Luo et al., 2022)
• "Entropies of commuting transformations on Hilbert spaces" (Li et al., 2020)
• "On the entropy of conservative flows" (Bessa et al., 2010)