Relative Topological Entropy

• Relative topological entropy is defined as the exponential growth rate of distinguishable orbit segments in fibers over a given factor, quantifying additional complexity in dynamical systems.
• It extends classical topological entropy through weighted generalizations and mean dimension frameworks, incorporating variational principles and averaging techniques for amenable groups.
• The invariant is crucial for distinguishing orbit complexity in extensions, aiding in the classification of systems with zero or infinite fiber complexity.

Relative topological entropy is a central invariant in topological dynamics, measuring the orbit complexity of a dynamical system relative to a chosen factor or subsystem. It generalizes classical topological entropy by quantifying the exponential growth rate of distinguishable orbit segments in the fibers above the factor, and further extends to weighted and mean dimension settings to address more subtle forms of complexity. Variational principles, covering and spanning structures, and group-theoretic averaging play integral roles in its definition and analysis, particularly for actions of amenable groups and factor maps.

1. Formal Definitions and Characterizations

Relative topological entropy is most commonly formalized for a factor map $\pi : X \to Y$ between compact topological dynamical systems and for group actions $(X, G)$ via homeomorphisms. Given a finite open cover $\alpha$ of $X$ and a Følner sequence $(F_n)$ in a countable discrete amenable group $G$, the open-cover definition is

$$H(\alpha; \pi) = \sup_{y \in Y} \limsup_{n \to \infty} \frac{1}{|F_n|} \log N(\alpha^{F_n}, \pi^{-1}(y)),$$

where $N(\alpha^{F_n}, \pi^{-1}(y))$ is the smallest cardinality of a subcover of the fiber $\pi^{-1}(y)$ by sets from the joint cover $\alpha^{F_n}$. The supremum over $\alpha$ yields the relative topological entropy $h_{\mathrm{top}}(\pi ; X | Y)$ (Liu et al., 22 Nov 2025).

An equivalent formulation—using metrics associated to the system—relies on $(F_n)$ and the induced metrics $d_{F_n}$ on $X$ fibers, considering the maximal cardinality of $\epsilon$-separated sets. Independence from the specific Følner sequence and, in the case of non-discrete amenable groups, independence from the choice of averaging net (such as Van Hove nets) is guaranteed by results analogous to the Ornstein–Weiss lemma (Hauser, 2019).

Weighted generalizations specify a continuous potential $f \in C(X, \mathbb{R})$ and a weight $w \in [0,1]$. The relative $w$-weighted topological entropy $h^w_{\mathrm{top}}(X|Y, \pi)$ specializes from the relative weighted topological pressure $P_Z(T, \pi, f)$ when $f \equiv 0$ (Yin, 2024).

2. Measure-theoretic and Variational Principles

A fundamental aspect of relative topological entropy is its connection to invariant probability measures and conditional entropy. For three compact metric systems $(X, T)$, $(Y, S)$, $(Z, R)$ with factor maps $\pi: X \to Y$, $\sigma: Y \to Z$, and composite $\rho = \sigma \circ \pi$, the conditional entropy of $\mu \in M(X, T)$ above $R$ is given by

$$h_\mu(T | R) = h_\mu(T | \sigma^{-1} \mathcal{B}_Z) = h_\mu(T | \pi^{-1} \mathcal{B}_Y ) + h_{\pi_+ \mu}(S | \sigma^{-1} \mathcal{B}_Z ),$$

where $\mathcal{B}_Y$, $\mathcal{B}_Z$ denote Borel $\sigma$-algebras (Yin, 2024).

The main variational principle for relative $w$-weighted pressure asserts: $$P_Z(T, \pi, f) = \sup_{\mu \in M(X, T)} \left[w h_\mu(T|R) + (1-w) h_{\pi_+ \mu}(S|R) + w \int f d\mu \right].$$ Specializing to $w=1$ recovers Ledrappier–Walters’ classical principle of relative entropy and pressure (Yin, 2024).

3. Extensions to Amenable Groups and Non-discrete Actions

Relative topological entropy has been extensively studied for actions of non-discrete amenable groups on compact spaces. The construction utilizes uniformities (entourages), Van Hove nets, and Haar measures. Given a factor map $\pi: X \to Y$ and a group $G$ acting on $X$, the relative entropy (using Van Hove nets $\{F_i\}$) is

$$h_{\mathrm{top}}(G,X|Y) = \sup_{\eta \in \mathcal{U}_X } \limsup_{i \to \infty} \frac{1}{\mu(F_i)} \log \left[ \mathrm{Cov}_\pi (N_\eta(F_i)) \right ]$$

where $N_\eta(F_i)$ is the Bowen-entourage and $\mathrm{Cov}_\pi(\cdot)$ is the maximal covering number taken over fibers (Hauser, 2019).

This formalism extends entropy-addition results (Bowen’s formula), and invariance under choice of averaging net, providing a unified framework over both discrete and continuum groups. The independence of Van Hove nets is established for locally compact amenable groups and is essential for applications to aperiodic order and cut-and-project schemes (Hauser, 2019).

4. Relative Mean Dimension and Entropy Dimension

Relative mean dimension $mdim(\pi; X|Y)$ measures the orbit complexity in terms of real parameters per group element, complementing the exponential bit-complexity of entropy. For $\pi: X \to Y$, metric $d$ on $X$, and Følner sequence $(F_n)$,

$$mdim(\pi; X|Y) = \lim_{\epsilon \to 0}\lim_{n \to \infty} \frac{1}{|F_n|} \sup_{y \in Y} widim_\epsilon ( \pi^{-1}(y), d_{F_n} ),$$

where $widim_\epsilon$ is the minimal $k$ such that the fiber admits an $\epsilon$-embedding into a $k$-dimensional simplicial complex (Liu et al., 22 Nov 2025).

An analogous notion is the relative entropy dimension $D(X,G|\pi)$, defined via critical exponents describing the sub-exponential growth rates of covering numbers: $$D_{\sup}(X,G|\pi) = \sup_{\mathcal{U}}\inf\{a \ge 0 : h_{\sup}(G,\mathcal{U}, a|\pi) = 0\}$$ and similar for $D_{\inf}$, utilizing refinements over all finite open covers (Xiao et al., 2022). These dimensions coincide with those derived from entropy-generating sets and reflect the full spectrum of fiber complexity below classical entropy scale.

5. Induced Factors, Probability Measures, and Disjointness

For a factor map $\pi: (X,G) \to (Y,G)$, the induced factor on spaces of probability measures $\widetilde{\pi}: (M(X),G) \to (M(Y),G)$ preserves and reflects relative entropy properties. Key theorems establish that:

• $h_{\mathrm{top}}(\pi; X | Y) = 0$ if and only if $h_{\mathrm{top}}(\widetilde{\pi}; M(X) | M(Y)) = 0$
• $h_{\mathrm{top}}(\pi; X | Y) > 0$ if and only if $mdim(\widetilde{\pi}) = \infty$

These equivalences utilize combinatorial independence sets and covering-dimension arguments, notably the Karpovsky–Milman lemma for binary patterns and Lebesgue-covering lemmas for simplices (Liu et al., 22 Nov 2025).

Relative entropy dimensions also govern disjointness properties of extensions: if the $n$th relative dimension set $\mathcal{D}_n(X,G|\pi_X)$ for one system strictly dominates that for another system over a common factor $Z$, then the systems are disjoint over $Z$, i.e., admit no nontrivial joining (Xiao et al., 2022).

6. Classical Recovery, Weighted Formalism, and Limit Cases

When the weight $w=1$ in the relative weighted pressure formalism, or when factors are trivial (e.g., $Z$ is a point), all relative notions collapse to classical absolute definitions: Ledrappier–Walters’ principle for entropy and pressure,

$$P(T, \pi, f) = \sup_{\mu: \pi_+ \mu = \nu} [h_\mu(T|\pi) + \int f d\mu]$$

Further, when the factor is trivial, entropy dimensions and mean dimension become absolute invariants for the system $(X,G)$, subsuming earlier results of Dou, Huang, and Park for $\mathbb{Z}$-actions (Xiao et al., 2022).

Weighted approaches, as formalized by Tsukamoto and extended by Yin, permit flexible thermodynamic formalism and enable interpolation between purely topological and measure-theoretic complexity contributions (Yin, 2024).

7. Significance and Applications

Relative topological entropy serves as a foundational tool for distinguishing orbit complexity in extensions, quantifying how much additional information is retained in fibers of a factor map beyond the factor’s own dynamics. Its weighted and dimension-theoretic variants address finer distinctions of complexity relevant to zero-entropy systems, sub-exponential growth, and non-classical settings such as cut-and-project schemes or actions of non-discrete amenable groups.

The connections to mean dimension and induced factors have far-reaching implications for problems in ergodic theory, symbolic dynamics, aperiodic order, and topological classification. The equivalence between positive relative entropy and infinite mean dimension of induced factors unifies combinatorial and geometric perspectives in the study of orbit growth rates (Liu et al., 22 Nov 2025).

The independence of entropy from averaging procedures (Følner or Van Hove sequences) and the ability to reduce non-discrete group actions to discrete lattice cases greatly enhance the generality and applicability of the theory (Hauser, 2019). The relative dimension set framework provides robust criteria for dynamical disjointness, extending classical joining results to amenable-group actions and complex extensions (Xiao et al., 2022).

A plausible implication is that these theories enable precise characterizations of extensions with zero or infinite fiber complexity, laying foundations for further classification and rigidity phenomena in dynamical systems.