Intermediate Topological Pressures

• Intermediate topological pressures are invariants that interpolate between global topological measures and local subset-based quantities in dynamical systems and condensed matter.
• They are constructed using parametrized Carathéodory methods, providing a continuous spectrum that bridges classical pressures with capacity and multifractal analysis.
• In condensed matter, these pressures identify critical points for electronic and topological phase transitions, correlating Fermi surface reconnections with changes in Z2 invariants.

Intermediate topological pressures formalize a spectrum of invariants—both in dynamical systems and condensed matter contexts—quantifying complexity or phase structure that interpolates between traditional (global) topological quantities and their finer, local, or subset-supported analogs. In dynamical systems, these include pressures defined via Carathéodory-type constructions, pressures on level sets or invariant subsets, and parametrized families that interpolate between Hausdorff- and capacity-type quantities. In electronic structure theory and materials science, "intermediate topological pressures" refer to sequences of critical external pressures at which electronic or topological phase transitions and Fermi surface reconnections occur, separating distinct topological regimes. The joint mathematical and physical perspectives yield a comprehensive framework for analyzing multifractal spectra, variational principles, dimension theory, and symmetry-protected topological changes as a function of system or external parameters.

1. Parametrized Families of Intermediate Topological Pressures

A principal approach to intermediate topological pressures employs a one-parameter family indexed by $\theta\in[0,1]$, providing an interpolation between classical Pesin–Pitskel (Bowen) pressures and capacity/type pressures, notably in both autonomous and nonautonomous dynamical systems. Given a (possibly nonautonomous) sequence of maps $\{f_j\}$ on a compact metric space $X$ and continuous potential $\varphi\in C(X)$, the $\theta$-intermediate pressure is defined through a Carathéodory–Pesin covering structure, restricting the lengths $m$ of admissible strings in the cover by

$N \le m < N/\theta + 1.$

When $\theta=0$, the condition $N \le m<\infty$ recovers the classical Pesin–Pitskel pressure $P^B(f,Z,\varphi)$, while $\theta=1$ constrains $m=N$, producing lower and upper capacity pressures corresponding to box dimensions. For $0<\theta<1$ this construction interpolates, yielding a continuous, nondecreasing curve of intermediate pressures in $(0,1]$. These pressures are monotone in $\theta$, admit a power rule ($P(f^m,Z,S_m\varphi,\theta)=m P(f,Z,\varphi,\theta)$), and are continuous in both $\varphi$ and, provided $\theta>0$, in $\theta$ as well. They satisfy factor map inequalities and, under appropriate regularity, allow for a variational principle relating the topological and measure-theoretic versions of intermediate pressure (Ju, 1 Jan 2026).

2. Intermediate Pressures on Subsets and Level Sets

The extension of topological pressure to non-compact subsets ("intermediate" between global and local) is formalized via Bowen's non-compact pressure and further refined through Carathéodory-type constructions on analytic sets, compact subsets, and level sets. For a given continuous map $f:X\to X$, potential $f$, and compact $K\subset X$, the Bowen pressure on $K$ is defined by weighted covers of $K$ by Bowen balls, leading to a critical exponent $P_B(T,f,K)$ that generalizes global pressure. This quantity interpolates between the global topological pressure (when $K=X$) and measure-theoretic pressure (when $K$ is small or even a single point supported by a measure). The variational principle holds:

$P_B(T,f,K)=\sup\bigl\{P_\mu(T,f):\mu(K)=1\bigr\},$

where $P_\mu(T,f)$ is the measure-theoretic pressure. Analytic subsets are handled by supremizing over their compact subsets (Tang et al., 2013). These intermediate set pressures provide the foundation for multifractal analysis, as exemplified by pressure functions on simultaneous level sets $$E(\alpha)=\{x:\lim_{n\to\infty}S_n\varphi(x)/S_n\psi(x)=\alpha\}$$, where conditional variational principles and Legendre transforms yield fine spectra and dimension-theoretic results (Climenhaga, 2011).

3. Intermediate Pressures in Nonautonomous and Irregular Structures

The extension to nonautonomous dynamical systems introduces a hierarchy of six "intermediate" pressures—spanning/packing, lower/upper, and Carathéodory-Bowen/Pesin-Pitskel types—which mirror the classical dimension sequence (Hausdorff, lower and upper box, packing) in fractal geometry (Chen et al., 2 Aug 2025). These pressures capture complexities in systems without time invariance and are crucial when constructing variational principles, power and product rules. For instance, under equicontinuity and equiconjugacy, the pressure hierarchy collapses, but for nonregular subsystems (e.g., Moran-like constructions), inequalities in the capacity chain persist, with intermediate pressures $\underline{P},\overline{P}$ characterizing degrees of dynamical complexity.

4. Dense Realization of Intermediate Pressures: Climenhaga–Thompson Structures

Intermediate pressures, in the sense of ergodic measure realizations, are made precise in systems with the Climenhaga–Thompson (CT) structure (Sun, 2019). Here, for $(X,f,\varphi)$ admitting such a structure, the set of realized ergodic pressures $P^e(X,f,\varphi)$ is dense (in fact fills) the interval $[P^*(\varphi),P(f,\varphi)]$, where $$P^*(\varphi) = \liminf_{n\to\infty}\frac1n \sup_{x\in X}S_n\varphi(x)$$ is the lower topological-potential bound. The decomposition into "good", "poor", and "small" orbit-segments, together with orbit gluing and Bowen regularity, enables the construction of invariant sets and measures with prescribed pressure, thereby yielding a continuum of intermediate values. This has direct implications for the metric entropy spectrum in Mañé diffeomorphisms, supporting every value in $[0,h_{\text{top}}(g)]$ (Sun, 2019).

5. Multifaceted Roles of Intermediate Topological Pressures

Intermediate topological pressures serve as bridges between topological, ergodic, and dimension-theoretic quantities:

• In the multifractal analysis of dynamical systems, pressures on level sets of Birkhoff ratios or Lyapunov exponents enable the retrieval of entropy and dimension spectra via normalized Legendre transforms, subject to conditional variational principles under thermodynamic regularity (Climenhaga, 2011).
• For stable and unstable sets in positive entropy systems, the $\epsilon$–stable-preimage pressures localize complexity, interpolating between the measure-theoretic and global pressures. For almost every $x$, the local pressure $P_s(T,\varphi,x,\epsilon)$ satisfies

$$\liminf_{\epsilon\to0} P_s(T,\varphi,x,\epsilon) \ge h_\mu(T) + \int\varphi\,d\mu,$$

and the global topological pressure is recovered as the $\epsilon\to0$ limit of the supremum over $x$ (Ma et al., 2010).

6. Intermediate Topological Pressures in Condensed Matter: Critical Pressures for Topological Transitions

In electronic structure theory, the term "intermediate topological pressures" most commonly designates a series of critical hydrostatic pressures $P_{ci}$ at which a material undergoes topological quantum phase transitions (TQPTs) or Lifshitz (electronic topological) transitions. Upon varying pressure, these transitions correspond to:

• Fermi surface topology changes (e.g., formation/collapse of pockets or necks), as in elemental Cd, which exhibits five distinct ETTs at $P_{c1}=2$ GPa, $P_{c2}=8$ GPa, $P_{c3}=12$ GPa, $P_{c4}=18$ GPa, $P_{c5}=28$ GPa, each with associated anomalies in elastic constants, lattice parameters, and compressibility (Srinivasan et al., 2015).
• Changes in $Z_2$ topological invariants marking insulator-to-trivial or nontrivial phase boundaries (e.g. in ZnGeSb$_2$, where a discontinuous tetragonal distortion at $P_c\approx6.8$ GPa marks the transition from strong topological insulator to trivial insulator; $Z_2$ switches from $(1;000)$ to $(0;000)$) (Sadhukhan et al., 2022).
• Closure and reopening of the bulk gap with band inversion, as captured by pressure-dependent Lüttinger or $k\cdot p$ models (e.g. CdSnSb$_2$ and CdGeSb$_2$ exhibit a single $P_c\approx2.2$–$2.3$ GPa at which the system passes from 3D topological insulator to Dirac semimetal to trivial insulator, with the Dirac cone only at the gap-closing point) (Juneja et al., 2018).

These electronic intermediate pressures demarcate a sequence of topologically distinct regimes—often summarized as TI/DS/NI phases—observable via DFT calculations, analysis of $Z_2$ (via Wilson-loop methods), and critical discontinuities in structural order parameters.

7. Connections to Fractal Dimensions and Practical Implications

There is a sharp analogy between the hierarchy of intermediate pressures and the sequence of fractal dimensions (Hausdorff, box, packing). Under various regularity or dynamical invariance assumptions, these pressures collapse or strictly separate, reflecting the underlying geometric or dynamical irregularity. The product and scaling laws for pressures match those of the corresponding dimension theory (Chen et al., 2 Aug 2025). In practical analysis—dynamical, multifractal, or material—the diversity of intermediate pressures provides both theoretical and computational tools for quantifying transitions, complexity, and the fine structure of invariant sets, spectral properties, and physical response at critical regimes.

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References:

• "Topological pressure of simultaneous level sets" (Climenhaga, 2011)
• "Variational principles for topological pressures on subsets" (Tang et al., 2013)
• "Electronic topological transitions in Cd at high pressures" (Srinivasan et al., 2015)
• "Intermediate topological presssures and variational principles for nonautonomous dynamical systems" (Ju, 1 Jan 2026)
• "Topological pressures for $ε$-stable and stable sets" (Ma et al., 2010)
• "Denseness of intermediate pressures for systems with the Climenhaga-Thompson structures" (Sun, 2019)
• "Pressure driven topological phase transition in chalcopyrite ZnGeSb$_2$" (Sadhukhan et al., 2022)
• "Nonautonomous Dynamical Systems I: Topological Pressures and Entropies" (Chen et al., 2 Aug 2025)
• "Pressure-Induced Topological Phase Transitions in CdGeSb$_2$ and CdSnSb$_2$" (Juneja et al., 2018)