Intermediate Measure-Theoretic Pressures

• Intermediate measure-theoretic pressures are quantitative invariants that interpolate between classical topological pressure and measure-theoretic entropy.
• They are defined through Carathéodory-type coverings, separated set methods, and local growth rates that link dimension theory with thermodynamic formalism.
• Their framework extends to nonautonomous, noncompact, and group settings, offering refined variational principles and multifractal insights.

Intermediate measure-theoretic pressures are quantitative invariants in topological and measurable dynamical systems that interpolate between classical notions of topological pressure and measure-theoretic entropy. They serve to refine the standard variational principles and enable the analysis of systems with non-ergodic measures, time-dependent (nonautonomous) dynamics, noncompact phase spaces, or multifractal properties. These pressures arise as critical exponents in Carathéodory-type coverings, separated-set constructions, or as upper and lower limits of local growth rates for measures of dynamical “balls,” and are tightly linked to dimension theory (e.g., multifractal spectra, Hausdorff dimension), thermodynamic formalism, and the structure of the pressure spectrum in both uniform and non-uniformly hyperbolic systems.

1. Foundational Definitions and Central Constructions

The concept of intermediate measure-theoretic pressure generalizes classical pressure notions by allowing for Carathéodory-Pesin, separated, or localized (packing/Bowen) formulations. Given a compact metric space $(X,d)$, a continuous map $f:X\to X$, a continuous potential $\varphi\in C(X,\mathbb{R})$, and a $f$-invariant Borel probability measure $\mu$, the intermediate measure-theoretic pressure may be defined via:

• Carathéodory-Pesin Structure: For a subset $Z\subset X$, potential sequence $\Phi=\{\phi_n\}$ with $\phi_n(x)=\sum_{i=0}^{n-1}\varphi(f^i x)$, and parameters $a\in\mathbb{R}$, $N\in\mathbb{N}$, $\epsilon>0$, set

$$M(Z, \Phi, a, N, \epsilon) = \inf\left\{\sum_{i} \exp(-a n_i + \phi_{n_i}(x_i)) : Z \subset \bigcup_i B_{n_i}(x_i, \epsilon), n_i \ge N\right\}$$

The measure-theoretic pressure $P_C(\mu, \Phi)$ is then the limit over $\epsilon \to 0$ and measure-class coverings $Z$ with $\mu(Z)=1$.

• Separated Set Approach: For weak-$*$ neighborhood $F$ of $\mu$, consider the empirical set $X_{n,F}$ and $(n,\epsilon)$-separated subsets $S\subset X_{n,F}$, defining

$$P(F; \Phi, n, \epsilon) = \sup_{S\subset X_{n,F} \;\text{sep.}} \sum_{x\in S} \exp(\phi_n(x))$$

The pressures $\underline{P}_S$, $\overline{P}_S$ are then infima over $F$ of $\liminf$ and $\limsup$ growth rates.

• Local and Packing/Bowen Pressures: For $x\in X$, define

$$P^+_\mu(f, x) = \lim_{\epsilon \to 0} \limsup_{n \to \infty} \frac{-\log\mu(B_n(x, \epsilon)) + \phi_n(x)}{n}$$

and integrate this over $\mu$ to obtain global pressure quantities (Zhong et al., 2022).

• Parameterized Families: In nonautonomous or non-standard settings, admit a covering parameter $\theta \in [0,1]$ controlling the admissible lengths in the Carathéodory construction. The lower and upper $\theta$-intermediate measure-theoretic pressures, $\underline P_\mu(\boldsymbol{f}, \varphi, \theta)$ and $\overline P_\mu(\boldsymbol{f}, \varphi, \theta)$, interpolate between Bowen–Pesin–Pitskel and capacity pressures (Ju, 1 Jan 2026).

The ergodic case universally recovers the classical free energy: $P_\mu(\varphi) = h_\mu(f) + \int \varphi \, d\mu$ In the non-ergodic setting, more delicate variational and supremum constructions are necessary (Fang et al., 2019).

2. Variational Principles, Billingsley-type Theorems, and Spectra

For a dynamical system and given potential, the relation between topological, packing, and measure-theoretic pressures is governed by variational principles:

• Bowen/Pesin–Pitskel Pressure: For a compact $K\subset X$ under appropriate regularity,

$$P^B(K, f, \varphi) = \sup_{\mu(K)=1} \underline{P}_\mu(f, \varphi)$$

where $\underline{P}_\mu$ is the lower global measure-theoretic pressure.

• Packing Pressure: If $P^P(K, f, \varphi) > \|\varphi\|_\infty$,

$$P^P(K, f, \varphi) = \sup_{\mu(K)=1} \overline{P}_\mu(f, \varphi)$$

where $\overline{P}_\mu$ is the upper pressure (Zhong et al., 2022, Xiao et al., 2024, Chen et al., 28 Feb 2025).

• Billingsley-type Theorems: For $s\in\mathbb{R}$ and Borel $Z$:
  • If $P^+_\mu(x) \le s$ for all $x \in Z$, then $P^P(Z, T, f) \le s$
  • If $P^+_\mu(x) \ge s$ for all $x\in Z$ and $\mu(Z)>0$, then $P^P(Z, T, f) \ge s$
  • This characterizes topological (or packing) pressure by pointwise upper local pressure (Zhong et al., 2022, Xiao et al., 2024, Chen et al., 28 Feb 2025).
• Nonautonomous Systems: The lower and upper global intermediate pressures relate to topological (Bowen, packing) pressures via supremums over measure-theoretic lower and upper pressures. These variational principles extend to time-dependent systems and provide explicit connection between dimension-theoretic and thermodynamic invariants (Chen et al., 28 Feb 2025, Ju, 1 Jan 2026).

3. Structure, Density, and Interpolation of Pressure Spectra

Intermediate measure-theoretic pressures are fundamental in describing the possible values for $h_\mu(f) + \int \varphi \, d\mu$ as $\mu$ runs over ergodic measures, yielding the so-called pressure spectrum. Structural results include:

• Interval Filling and Density: In systems admitting the intermediate pressure property, the spectrum of ergodic measure pressures fills the whole interval $[P_{\inf}, P_{\sup}]$:

$$\{P_\mu(f, \varphi) : \mu \in \mathcal{M}_{\rm erg}(f)\} = [P_{\inf}(f, \varphi), P_{\rm top}(f, \varphi)]$$

This holds generically for geometric Lorenz and singular hyperbolic attractors, as established using gluing-orbit and specification properties (Shi et al., 2024, Sun, 2019).

• Gaps in the Spectrum: For dense sets of non-generic systems, notably in geometric Lorenz attractors with specific observables, genuine intervals may be missing from the attainable spectrum of measure-theoretic pressures. Typical mechanism involves isolated ergodic measures (e.g., Dirac masses at singularities) with abrupt pressure “cliffs” (Shi et al., 2024).
• Denseness Results: In systems with Climenhaga–Thompson structure, the set of ergodic measure pressures is dense in $[P_*(\varphi), P(\varphi)]$, where $P_*(\varphi)$ is the liminf supremum of Birkhoff sums. This interpolates not only for pressure but also entropy (Katok's intermediate-entropy theorem) (Sun, 2019).

4. Intermediate Pressures in Nonautonomous, Noncompact, and Group Contexts

The formalism extends beyond classical autonomous compact systems:

• Nonautonomous Settings: Intermediate measure-theoretic pressures are defined for sequences of maps $(T_k)$ and equicontinuous potentials, yielding local and global lower/upper pressures $\underline P_\mu, \overline P_\mu$ and variational principles for Bowen/packing pressures on arbitrary compact subsets (Chen et al., 28 Feb 2025, Ju, 1 Jan 2026).
• Parameter Interpolation: The one-parameter family $P_\mu^\theta$ introduced in (Ju, 1 Jan 2026) interpolates between the Pesin–Pitskel and upper/lower capacity pressures, with monotonicity and continuity in $\theta$, and power and factor rules. The limiting cases yield the classical variational inequalities and equalities.
• Amenable Groups and Noncompact Spaces: The measure-theoretic pressures extend to amenable group actions through (scaled) Bowen/packing formulations, giving Billingsley-type and variational results in this higher generality (Xiao et al., 2024, Briceño, 2018). For locally compact (possibly noncompact) spaces, admissible cover techniques ensure the validity of a variational principle, avoiding “entropy/pressure leakage” from infinity (Caldas et al., 2021).

5. Connections to Entropy, Dimension Theory, and Multifractal Analysis

Intermediate measure-theoretic pressures enable refined variational characterizations and dimension formulas:

• Entropy and Free Energy: For ergodic measures, multiple constructions of measure-theoretic pressure coincide with $h_\mu(f) + \int \varphi\, d\mu$ (Zhong et al., 2022). For non-ergodic measures, the pressure is the essential supremum over ergodic decompositions (Fang et al., 2019).
• Dimension Formulas: On average-conformal repellers and hyperbolic sets, the Hausdorff dimension of a measure is the unique zero of the pressure function $t \mapsto P_\mu(-t\varphi)$ (Fang et al., 2019). Similar formulas hold for dynamical systems and repeller sets with non-uniform or partial hyperbolicity (Chen et al., 28 Feb 2025).
• Multifractal Formalism: The fine structure of the pressure spectrum, especially gaps or density, is essential for multifractal decompositions, zero-temperature limits, and the geometry of invariant measures (Shi et al., 2024, Sun, 2019).

6. Further Developments and Open Problems

Recent works have generalized intermediate measure-theoretic pressure theory in several directions:

• Nonautonomous and Parameterized Pressures: Systematic study of intermediate families parameterized by $\theta$ reveals new regularity and continuity phenomena, together with variational principles unifying classical extremes (Ju, 1 Jan 2026).
• Structural Dichotomies: The existence of systems where the intermediate-value property of the pressure spectrum fails (as in geometric Lorenz attractors) shows the necessity of understanding obstructions, such as non-isolation of ergodic measures or lack of specification (Shi et al., 2024).
• Group Actions and Amenable Cases: Recent advances provide SMB-type representations and variational formulas for intermediate pressures in amenable group actions and virtually orderable groups (Briceño, 2018), connecting pointwise information convergence to global pressure representation.

7. Summary Table: Core Constructions and Variational Principles

Formulation	Main Definition/Construction	Variational Principle
Carathéodory-Pesin	Critical exponent for infimum of weighted coverings of $\mu$-full sets	$$P_C(\mu, \Phi) = \lim_{\epsilon\to 0} \inf_{Z: \mu(Z)=1} P_Z(f, \Phi, \epsilon)$$
Separated Set	Supremum over separated sets of empirical measures & potential sums	$$P_S(\mu, \Phi) = \lim_{\epsilon\to 0} \inf_{F \ni \mu} \limsup_{n\to \infty} (1/n) \log P(F)$$
Packing/Bowen Local	Upper/lower local growth rates of $\mu$-measure of dynamical balls	$$P^P(Z) = \sup\{ \overline{P}_\mu(T, f): \mu(Z)=1 \}$$ ($\|\varphi\|_\infty$ bounded away)
Parameterized ($\theta$)	Covers with lengths in $[N, N/\theta + 1)$; interpolate between pressure types	$$P_{\text{top}}^\theta = \sup_\mu \underline P_\mu(\theta)$$; recovers classical cases at limits

These constructions characterize a robust and flexible framework for studying complexity, dimension, and thermodynamic properties in broad classes of dynamical systems, unifying classical, multifractal, and nonautonomous phenomena (Fang et al., 2019, Zhong et al., 2022, Shi et al., 2024, Ju, 1 Jan 2026).