Carathéodory-Pesin Structure Overview

• Carathéodory-Pesin structure is a framework that unifies thermodynamic invariants, such as pressure and entropy, with geometric measure theory in dynamical systems.
• It employs leaf-wise constructions, variable-length coverings, and intermediate pressures to extend classical notions to noncompact, non-invariant, and nonautonomous settings.
• The theory facilitates multifractal analysis and rigorous Hausdorff dimension computations while linking traditional dynamical systems analysis with contemporary ergodic theory.

The Carathéodory–Pesin structure (abbreviated “C–P structure”; Editor's term) provides a foundational framework for encoding and generalizing dimensional characteristics and thermodynamic invariants—such as topological pressure, entropy, and equilibrium measures—within dynamical systems, particularly addressing arbitrary, non-invariant, and non-compact subsets. Developed in the context of both discrete-time maps and continuous flows, the C–P structure synthesizes geometric measure theory and thermodynamic formalism. Key advances include leaf-wise constructions for hyperbolic systems, extensions to semigroup actions and nonautonomous systems via variable-length coverings, and the introduction of a one-parameter family of intermediate pressures interpolating between classical notions. The theory applies equally to compact, noncompact, and locally compact phase spaces, and furnishes rigorous tools for multifractal analysis and Hausdorff dimension computations (Climenhaga, 2020, Xiao et al., 2020, Ju, 1 Jan 2026, Ma et al., 2016).

1. Formal Definition of the Carathéodory–Pesin Structure

Let $(X,d)$ be a metric space (compact or locally compact) and $f:X\to X$ a continuous transformation, or more generally, a family of maps from a free semigroup or a nonautonomous sequence. Given a continuous potential function $\phi:X\to\mathbb R$, the construction proceeds by covering an arbitrary set $E\subset X$ with dynamical balls or cylinders. For a fixed scale $\varepsilon>0$, the (forward) Bowen ball of order $n$ at $x$ is

$$B_n(x,\varepsilon) = \{ y \in X : d(f^k y, f^k x) < \varepsilon,\;\forall 0\le k < n \}.$$

For each real $s$, define the weighted outer Carathéodory–Pesin measure via

$$\mathcal{M}_\phi^s(E,N) = \inf \left\{\sum_i e^{-s n_i + S_{n_i} \phi(x_i)} \;\middle|\; E \subset \bigcup_i B_{n_i}(x_i, \varepsilon), n_i \ge N \right\},$$

where $S_{n} \phi(x) = \sum_{k=0}^{n-1} \phi(f^k x)$. Passing to the limit in $N$ yields

$$\mathcal{M}_\phi^s(E) = \lim_{N\to\infty} \mathcal{M}_\phi^s(E, N).$$

The critical exponent $P(\phi)$ is defined as the value where $\mathcal{M}_\phi^s(X)$ jumps from $+\infty$ to $0$. This captures the topological pressure through several equivalent formulations: $$P(\phi) = \inf\{ s : \mathcal{M}_\phi^s(X)=0 \} = \sup\{ s : \mathcal{M}_\phi^s(X) = \infty \}.$$ This definition recovers the Pesin–Pitskel pressure and, for $\phi = 0$, Bowen's topological entropy (Climenhaga, 2020, Ma et al., 2016).

For proper maps on locally compact spaces, covers are taken by admissible open sets, and pressure can be defined using strings of sets with variable or fixed lengths, resulting in several distinct notions (see Section 3).

2. Leaf-Wise Measures, Product Structures, and SRB Equilibrium

In uniformly hyperbolic systems, particularly on basic sets for flows and diffeomorphisms, the C–P structure facilitates the construction of measures along local stable and unstable manifolds (“leaves”). For a local unstable manifold $W^u_{\text{loc}}(x)$, restricting the C–P construction leaf-wise yields conditional equilibrium measures: $$m^u_x(Z) = \lim_{N\to\infty} \inf\left\{ \sum_i e^{-n_i P(\phi) + S_{n_i} \phi(x_i)} : Z \subset \bigcup_i B^u_{n_i}(x_i, \varepsilon) \right\},$$ with analogous formulae for stable leaves in backward time. These leaf-wise measures have strict scaling properties: $$m^u_x(B^u_n(x, \varepsilon)) = \exp(-n P(\phi) + S_n \phi(x)).$$ They satisfy positivity, conformality, and equivariance under dynamics: $$(f^t)_* m^u_x = e^{\Phi(x,t) - t P(\phi)} m^u_{f^t x},$$ where $\Phi(x,t)$ is the time-integrated potential. Patching these leaf measures via product structures on local rectangles

$$R = \{ [u,s] : u \in W^u_{\text{loc}}(x),\; s \in W^s_{\text{loc}}(x) \}$$

produces the unique global equilibrium state. The resulting measure displays the Gibbs property

$$C^{-1} \leq \frac{\mu(B_n(x, \varepsilon))}{\exp(S_n\phi(x) - n P(\phi))} \leq C,$$

and for the geometric potential, coincides with the SRB measure, whose conditionals are leaf-volume (Lebesgue) measures (Climenhaga, 2020).

3. Extensions: Semigroups, Nonautonomous Dynamics, and Intermediate Pressures

The C–P framework generalizes to free semigroup actions (generated by several maps $G_1 = \{ f_0, ..., f_{k-1} \}$), nonautonomous or time-dependent systems, and arbitrary subsets. For a vector-potential $\Phi = \{ \varphi_0, ..., \varphi_{k-1} \}$,

$$M(Z, G_1, \Phi, \alpha, \varepsilon, N) = \inf \sum_{j} \exp(-\alpha |w_j| + \sup_{y \in B_{w_j}(x_j, \varepsilon)} S_{w_j} \Phi(y)),$$

with the infimum running over covers by Bowen balls indexed by words $w_j$ of length at least $N$. The critical exponent in $\alpha$ yields the semigroup pressure.

Recent advances introduce a one-parameter family of “intermediate topological pressures”: $P(\boldsymbol{f}, Z, \varphi, \theta),$ where covers use strings of length $N \leq n < N/\theta + 1$ ($\theta \in [0,1]$), interpolating between classical Pesin–Pitskel pressure ($\theta = 0$) and lower/upper capacity pressures ($\theta=1$). All thermodynamic formalism properties—continuity, monotonicity, variational principle, factor-map inequalities—extend to this setting (Ju, 1 Jan 2026, Ma et al., 2016).

4. Variational Principles and Measure-Theoretic Pressures

For all C–P pressure notions, fundamental variational principles connect topological invariants to measure-theoretic entropy and integrals of potentials. On arbitrary subsets $Z$, or entire spaces, the supremum over invariant probability measures gives

$$P_Z(G_1, \Phi) = \sup \left\{ h_\mu(G_1) + \int \Phi\, d\mu : \mu \in \mathcal{M}_{G_1}(Z) \right\}.$$

Analogous statements hold for intermediate pressures and nonautonomous systems: $$P(\boldsymbol{f}, X, \varphi, \theta) = \sup_{\mu \in \mathcal{M}(X)} P_\mu(\boldsymbol{f}, \varphi, \theta),$$ where $P_\mu$ is defined via coverings using $\mu$-covers with the same intermediate constraints on lengths. These principles remain valid in noncompact, locally compact, and expansive systems and recover the classical full variational principle under specification (Ma et al., 2016, Ju, 1 Jan 2026).

5. Bowen’s Equation and Hausdorff Dimension

The C–P theory unifies the thermodynamic and dimensional analysis of (possibly noncompact) invariant sets. In the context of conformal semigroup actions, with expansion factors encoded by a geometric potential $\psi_i = \log a_i$, the C–P pressure admits a strictly decreasing family

$t \mapsto P_Z(G_1, -t \psi),$

with the unique root $t^*$ of $P_Z(G_1, -t^* \psi) = 0$ yielding the Hausdorff dimension: $$\dim_H Z = t^* = \sup\{ t : P_Z(G_1, -t \psi) > 0 \}.$$ In the case where all Lyapunov exponents are constant ($Z\subset A(\alpha)$), the dimension formula specializes to

$\dim_H Z = \frac{h_Z(G_1)}{\alpha}.$

This approach rigorously extends classical Bowen’s equation to noncompact, arbitrary subsets, under positivity of lower Lyapunov exponents and a “tempered contraction” condition essential for controlling growth rates (Xiao et al., 2020).

6. Classical and Contemporary Notions of Pressure

The C–P formalism incorporates several classical and modern pressure definitions. For proper maps, three distinct pressures can be defined: variable-length (“Pesin–Pitskel” type), lower capacity (fixed length, liminf), and upper capacity (fixed length, limsup). These coincide on compact, invariant sets but may differ for noncompact or non-invariant cases: $$P_Z(\varphi) \leq \underline{CP}_Z(\varphi) \leq \overline{CP}_Z(\varphi).$$ Capacity pressures can also be equivalently formulated via (n,$\varepsilon$)-separated or spanning sets. The framework reproduces and extends Patrão’s noncompact entropy and applies to multifractal analysis of local entropies, with spectra obtained by Legendre transforms of pressure functions (Ma et al., 2016).

7. Properties, Invariance, and Continuity

The Carathéodory–Pesin structure produces pressures and invariants with key properties:

• Monotonicity: $Z_1 \subset Z_2 \implies P_{Z_1} \leq P_{Z_2}$.
• Countable stability: $P_{\bigcup_i Z_i} = \sup_i P_{Z_i}$.
• Lipschitz continuity in potentials: $$|P_Z(\varphi) - P_Z(\psi)| \leq \|\varphi - \psi\|_\infty$$.
• Invariance under homeomorphism: $P_Z(\varphi) = P_{f(Z)}(\varphi)$.
• Conjugacy/factor-map inequalities for intermediate pressures: pressure does not increase under continuous surjection, with equality for homeomorphisms (Ju, 1 Jan 2026).
• Scaling (“power rule”): If the family of maps is equicontinuous, $$P(f^m, Z, S_m\varphi, \theta) = m P(f, Z, \varphi, \theta)$$.

These properties ensure robustness of the C–P formalism across a wide spectrum of dynamical and geometric contexts, providing a universal dimensional and thermodynamic toolkit for contemporary ergodic theory and dynamical systems analysis (Climenhaga, 2020, Xiao et al., 2020, Ju, 1 Jan 2026, Ma et al., 2016).