Induced Topological Pressure in Dynamical Systems

• Induced topological pressure is a pressure invariant in thermodynamic formalism that evaluates potentials against a weighted time scaling function and recovers classical pressure measures.
• It leverages inducing families, return-time truncations, and variational principles to determine equilibrium states and detect phase transitions in dynamical systems.
• The theory extends across countable, compact, and random settings, supporting nonlinear and high-dimensional generalizations with applications in multifractal analysis.

Searching arXiv for recent and foundational papers on induced topological pressure. Induced topological pressure is a pressure-like invariant in thermodynamic formalism in which a continuous or Hölder potential is evaluated relative to a positive “scaling” or “time-change” function, rather than against the unweighted iterate count used in classical topological pressure. In the countable-state setting, the notion was introduced for Markov shifts together with an arbitrary inducing family of finite words, and it was shown to recover classical pressure, Gurevič pressure, free-energy pseudo-inverses, and extensions of Savchenko’s entropy for special flows (Jaerisch et al., 2010). For compact topological dynamical systems, induced topological pressure was later formulated through return-time truncations determined by a positive continuous function $\psi$, and a variational principle identified it with a ratio of entropy-plus-potential to the $\psi$-average (Xing et al., 2015). More recent work studies equilibrium states, subdifferentials, freezing states, nonlinear high-dimensional versions, and random fiber analogues (Ma et al., 10 Jul 2025, Nan, 16 Apr 2026).

1. Foundational frameworks and basic objects

The original framework is a countable-state Markov shift $(\Sigma,\sigma)$ built from a countable alphabet $I\subset \mathbb N$ and an incidence matrix $A=(A_{ij})_{i,j\in I}$ with entries in $\{0,1\}$. One considers

$$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$

equipped with the left shift $\sigma$ and the metric

$$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$

where $|\omega\wedge\omega'|$ is the length of the longest common prefix. Finite admissible words form $\psi$0, and for $\psi$1 the cylinder $\psi$2 is the set of sequences with prefix $\psi$3 (Jaerisch et al., 2010).

In this setting one fixes two continuous functions,

$\psi$4

called respectively the potential and the scaling function, together with an arbitrary collection of finite words $\psi$5, called the inducing family. The data $\psi$6 determine the induced pressure. The role of $\psi$7 is to replace the usual iterate count by a weighted accumulation, while the role of $\psi$8 is to restrict the combinatorics to a chosen induced subsystem (Jaerisch et al., 2010).

For compact topological dynamical systems, the basic framework is a continuous map $\psi$9 on a compact metric space $(\Sigma,\sigma)$0, with $(\Sigma,\sigma)$1 and $(\Sigma,\sigma)$2. Here $(\Sigma,\sigma)$3 again acts as a time-change or scaling function. The construction uses Bowen metrics, Birkhoff sums $(\Sigma,\sigma)$4, and return-time truncations determined by the inequalities $(\Sigma,\sigma)$5 (Xing et al., 2015, Ma et al., 10 Jul 2025).

A further extension treats random dynamical systems. In that setting one starts with a Lebesgue base system $(\Sigma,\sigma)$6, a compact metric space $(\Sigma,\sigma)$7, a continuous bundle random dynamical system $(\Sigma,\sigma)$8, and $(\Sigma,\sigma)$9 with $I\subset \mathbb N$0 in the sense $I\subset \mathbb N$1 for $I\subset \mathbb N$2-a.e. $I\subset \mathbb N$3. The resulting object is a non-averaged induced fiber pressure defined through spanning or separated sets and averaged over the base via $I\subset \mathbb N$4 (Nan, 16 Apr 2026).

2. Definitions and pseudo-inverse characterizations

For countable-state Markov shifts, the induced partition sums are organized by $I\subset \mathbb N$5-accumulated time. For each integer $I\subset \mathbb N$6, one sums over those words $I\subset \mathbb N$7 satisfying the time constraint

$I\subset \mathbb N$8

for some, hence every, $I\subset \mathbb N$9, with weight $A=(A_{ij})_{i,j\in I}$0, where

$A=(A_{ij})_{i,j\in I}$1

The $A=(A_{ij})_{i,j\in I}$2-induced pressure of $A=(A_{ij})_{i,j\in I}$3 relative to $A=(A_{ij})_{i,j\in I}$4 is then

$A=(A_{ij})_{i,j\in I}$5

A basic theorem identifies this quantity with a critical exponent: $A=(A_{ij})_{i,j\in I}$6 This description is central because it makes the induced pressure a pseudo-inverse of an ordinary pressure problem (Jaerisch et al., 2010).

For compact systems, the return-time sets are

$A=(A_{ij})_{i,j\in I}$7

and

$A=(A_{ij})_{i,j\in I}$8

One then defines Bowen-style spanning and separated partition sums over the pieces $A=(A_{ij})_{i,j\in I}$9. In the 2025 formulation, the covering version is

$\{0,1\}$0

where each $\{0,1\}$1 is an $\{0,1\}$2-spanning set, and the separated version is defined analogously by a supremum over $\{0,1\}$3-separated sets. The induced topological pressure is

$\{0,1\}$4

and the spanning and separated constructions give the same number (Ma et al., 10 Jul 2025).

A decisive structural fact in both compact and random settings is the pseudo-inverse relation to classical pressure. In the compact case,

$\{0,1\}$5

where $\{0,1\}$6 is the usual topological pressure (Xing et al., 2015). In the random fiber setting, the corresponding statement is

$\{0,1\}$7

with $\{0,1\}$8 the classical fiber topological pressure (Nan, 16 Apr 2026).

This suggests a unifying interpretation: induced topological pressure is a root or critical exponent determined by the classical pressure of the tilted potential $\{0,1\}$9. That interpretation is explicit in the countable, compact, and random theories.

3. Variational principles, equilibrium states, and regularity

The variational principle in the compact deterministic setting states that

$$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$0

where $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$1 is the space of $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$2-invariant Borel probability measures and $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$3 is the measure-theoretic entropy. This formula is the main theorem of the compact theory introduced by Xing and Chen (Xing et al., 2015).

In the countable-state case, an analogous generalized Gurevič variational principle holds under irreducibility and Hölder assumptions. If $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$4 is irreducible, $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$5 are Hölder, and $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$6, then

$$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$7

which rearranges to the usual entropy-pressure-integral form (Jaerisch et al., 2010).

The 2025 deterministic theory develops equilibrium states for induced topological pressure. An equilibrium state for $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$8 is a measure $$\Sigma=\{\omega=(\omega_1,\omega_2,\ldots)\in I^{\mathbb N}:A_{\omega_n,\omega_{n+1}}=1\ \forall n\},$$9 attaining the supremum in the variational principle: $\sigma$0 The set $\sigma$1 of all such measures is convex, compact, and has ergodic extreme points. The same work defines the tangent functional, or subdifferential,

$\sigma$2

with $\sigma$3. If $\sigma$4 is Gateaux-differentiable at $\sigma$5, then $\sigma$6 is a singleton; otherwise non-differentiability signals a “first-order phase transition” (Ma et al., 10 Jul 2025).

The same paper also records standard pressure-type regularity. The map $\sigma$7 is convex and continuous in the uniform norm, and it is cohomology-invariant: if $\sigma$8, then $\sigma$9. It satisfies the bounds

$$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$0

In the countable Markov setting, corresponding properties include monotonicity in $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$1 and $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$2, convexity in $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$3 when $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$4, translation-invariance, subadditivity, and continuity under uniform-norm perturbations of $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$5 (Jaerisch et al., 2010, Ma et al., 10 Jul 2025).

4. Recovery of classical theories and induced constructions

Induced topological pressure is designed to recover several pre-existing pressure notions as special cases. In the countable-state Markov framework, if $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$6 and $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$7, then $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$8 agrees with the classical topological pressure for countable Markov shifts. If $$d_\alpha(\omega,\omega')=e^{-\alpha|\omega\wedge\omega'|},$$9 is mixing and $|\omega\wedge\omega'|$0, then $|\omega\wedge\omega'|$1 is exactly the Gurevič pressure of $|\omega\wedge\omega'|$2 (Jaerisch et al., 2010). In the compact setting, taking $|\omega\wedge\omega'|$3 recovers the usual topological pressure $|\omega\wedge\omega'|$4 (Xing et al., 2015, Ma et al., 10 Jul 2025).

The induced formalism also gives direct access to free-energy functions. In the countable-state theory, one has in many cases

$|\omega\wedge\omega'|$5

so that $|\omega\wedge\omega'|$6 is the “pseudo-inverse” or free-energy function of the $|\omega\wedge\omega'|$7-pressure (Jaerisch et al., 2010). In the compact theory, the same mechanism underlies the ratio variational principle through the map $|\omega\wedge\omega'|$8 (Xing et al., 2015).

A major application of the countable-state construction concerns special flows. Given a roof function $|\omega\wedge\omega'|$9, one forms the quotient

$\psi$00

with flow $\psi$01. For a continuous $\psi$02, let

$\psi$03

Under Hölder assumptions and the stated tail condition on $\psi$04, the paper proves

$\psi$05

in the irreducible case, thereby recovering and extending Savchenko’s formula for topological entropy, which corresponds to $\psi$06 (Jaerisch et al., 2010).

The same work introduces loop-space inducing and the exhausting principle. When the inducing family $\psi$07 has suitable concatenation and refinement properties, one can pass to a finite-type loop space $\psi$08 and obtain

$\psi$09

The exhausting principle requires an increasing sequence of compact $\psi$10-invariant sets $\psi$11 with dense union such that

$\psi$12

In dynamical group extensions, this becomes a criterion for amenability: if $\psi$13 is coded by a Markov shift of left-walks, then $\psi$14 is amenable if and only if the exhausting principle holds for $\psi$15 (Jaerisch et al., 2010).

5. Nonlinear, high-dimensional, and freezing extensions

The 2025 theory enlarges the classical ratio formalism to nonlinear high-dimensional induced pressure. Let $\psi$16 and $\psi$17 continuous. The nonlinear topological pressure without inducing is

$\psi$18

Under either of the standard hypotheses—abundance of ergodic measures or convexity of $\psi$19—one has

$\psi$20

The induced version $\psi$21 is obtained by replacing the fixed iterate $\psi$22 with the $\psi$23-truncated return-time decomposition $\psi$24, and it satisfies

$\psi$25

where $\psi$26 and $\psi$27 (Ma et al., 10 Jul 2025).

Several specializations are explicit. If $\psi$28 and $\psi$29, then $\psi$30. If $\psi$31, one recovers the high-dimensional nonlinear pressure $\psi$32. If $\psi$33 and $\psi$34, then $\psi$35 measures the “BS-dimension” where $\psi$36 is a gauge (Ma et al., 10 Jul 2025).

The same paper develops freezing states and zero-temperature limits for the family $\psi$37, $\psi$38. A measure $\psi$39 is a freezing state at temperature $\psi$40 if it is an equilibrium state for all $\psi$41. In that situation,

$\psi$42

becomes linear of slope

$\psi$43

and $\psi$44 is also the limit of $\psi$45-equilibria as $\psi$46, the “zero-temperature measure” (Ma et al., 10 Jul 2025).

A plausible implication is that induced pressure provides a common language for three operations that are often treated separately: changing the observable, changing the time scale, and applying a nonlinear functional to vector-valued Birkhoff averages. That unification is explicit in the formulas above.

6. Random fiber theory, examples, and terminological boundaries

For random dynamical systems, the non-averaged induced fiber pressure is defined through the induced-time set

$\psi$47

and the corresponding partition

$\psi$48

Using $\psi$49-spanning sets, one defines $\psi$50, and then

$\psi$51

Equivalent formulations via separated sets and open-cover analogues are available. The induced variational principle is

$\psi$52

where $\psi$53 denotes the $\psi$54-invariant probability measures on $\psi$55 projecting to $\psi$56 on $\psi$57, and $\psi$58 is the fiber entropy. The same paper extends the theory to nonlinear and higher-dimensional random pressures with the same critical-exponent and variational-principle structure (Nan, 16 Apr 2026).

Concrete examples already appear in the deterministic literature. For the one-sided full shift on $\psi$59 symbols with locally constant $\psi$60 and $\psi$61, one has

$\psi$62

so the induced pressure is the unique $\psi$63 satisfying

$\psi$64

In conformal iterated function systems, one can take $\psi$65, and then $\psi$66 is the generalized free-energy whose Legendre transform governs multifractal spectra. The renewal shift and the $\psi$67-Farey map give an explicit free-energy function $\psi$68 through the loop-space formalism (Xing et al., 2015, Jaerisch et al., 2010).

A terminological distinction is essential. In thermodynamic formalism, “induced topological pressure” refers to the pressure concepts described above. In several condensed-matter papers, by contrast, “pressure induced topological” refers to hydrostatic-pressure-driven topological quantum phase transitions in materials such as Sb$\psi$69Se$\psi$70, CdGeSb$\psi$71, CdSnSb$\psi$72, BiTeI, and Sb$\psi$73Te$\psi$74 (Li et al., 2013, Juneja et al., 2018, Qi et al., 2016, Zhu et al., 2013). Those works concern band inversion, $\psi$75 indices, Dirac surface states, and superconductivity under compression, not thermodynamic formalism.

Within dynamical systems, the mature picture is therefore stratified rather than singular: countable-state inducing emphasizes inducing families, loop spaces, and exhausting principles; compact deterministic theory emphasizes ratio variational principles, equilibrium states, and freezing; random theory emphasizes fiber entropy, non-averaged inducing, and higher-dimensional nonlinear extensions. Across these settings, the common invariant is the pressure determined by a potential $\psi$76 measured relative to a positive scaling function $\psi$77, with the decisive structural identity given by a pseudo-inverse or critical-exponent relation to classical topological pressure (Jaerisch et al., 2010, Xing et al., 2015, Ma et al., 10 Jul 2025, Nan, 16 Apr 2026).