Factor Maps for Amenable Group Actions

• Factor maps for amenable group actions are continuous, equivariant surjections that preserve key dynamical invariants, including entropy and topological pressure.
• The study reveals deep insights such as packing pressure inequalities, variational principles, and entropy rigidity, underscoring their role in transferring structural information.
• Applications span subshifts, algebraic extensions, and induced measure systems, highlighting their practical utility in analyzing the complexity of dynamical systems.

A factor map for amenable group actions provides a structural correspondence between dynamical systems endowed with actions of a countable, infinite, discrete amenable group. Factor maps preserve group equivariance and enable rigorous transfer and comparison of dynamical invariants—such as entropy, topological pressure, and conditional independence—across systems. The study of factor maps in this context has led to deep results on packing topological pressure, variational principles, entropy rigidity, and stability under extensions and induced maps, as documented in recent research on arXiv (Ding et al., 2024, Liu et al., 2022, Yin et al., 2023, Downarowicz et al., 2021, Frej et al., 2014).

1. Definitions: Amenable Groups, Dynamical Systems, and Factor Maps

Let $G$ be a countable infinite discrete amenable group. Amenability is characterized via a Følner sequence $\{F_n\}$, i.e., a sequence of finite subsets with $$\displaystyle\lim_{n\to\infty}\frac{|F_n\Delta gF_n|}{|F_n|}=0$$ for all $g\in G$. Temperedness of a Følner sequence requires a uniform upper bound on $|\bigcup_{k<n}F_k^{-1}F_n|/|F_n|$.

A $G$-action topological dynamical system $(X,G)$ comprises a compact metric space $X$ with $G$ acting by homeomorphisms. A factor map $\pi:(X,G)\to(Y,G)$ is a continuous surjection satisfying $\pi(g\cdot x)=g\cdot\pi(x)$ for all $g\in G$, $x\in X$. This equivariance ensures that dynamical structures relating to the group action are preserved.

2. Packing Topological Pressure and Factor Maps

Packing topological pressure, $P(Z,\{F_n\},f)$, generalizes entropy and pressure for amenable group actions. It is defined using Bowen metrics $d_{F_n}(x,y)=\max_{g\in F_n}d(g\cdot x,g\cdot y)$ and $(F_n,\epsilon)$-separated sets, aggregating weighted exponential counts over disjoint "Bowen balls."

The Ding–Chen–Zhou factor-map inequality (Ding et al., 2024) establishes that for any factor map $\pi:(X,G)\to(Y,G)$, tempered Følner sequence $\{F_n\}$, $f\in C(Y,\mathbb{R})$, and nonempty $E\subset X$,

• $P(\pi(E),\{F_n\},f) \leq P(E,\{F_n\},f\circ\pi)$,
• $$P(E,\{F_n\},f\circ\pi) \leq P(\pi(E),\{F_n\},f) + \sup_{y\in Y}h_{\mathrm{top}}(\pi^{-1}(y),\{F_n\})$$, where $h_{\mathrm{top}}(Z,\{F_n\})=P(Z,\{F_n\},0)$ is the packing entropy. The entropy of fibers quantifies the maximum loss or gain in pressure induced by the factor map.

3. Variational Principle and Measure-Theoretic Correspondence

Packing pressure on analytic sets admits a variational principle: $$P(Z,\{F_n\},f) = \sup_{\mu\in M(X,G),\ \mu(Z)=1} P_\mu(\{F_n\},f),$$ where $M(X,G)$ is the simplex of $G$-invariant Borel probability measures and $P_\mu$ is defined via Katok-style or local-pressure integrals. For $f\equiv 0$, one recovers the classical entropy variational principle.

Combining with the factor map inequality, the pressure of subsets under $\pi$ and their preimages relate via fiber-entropy corrections: $$\sup_{\mu\in M(X),\mu(Z)=1} P_\mu(\{F_n\},f\circ\pi) \geq P(Z,\{F_n\},f\circ\pi) - \sup_{y} h_{\mathrm{top}}(π^{-1}(y)).$$

4. Equality and Rigidity Under Additional Properties

Under the almost specification property or for ergodic invariant measures, equality is attained (Ding et al., 2024): $P(G_\mu,\{F_n\},f) = h_\mu(X,G) + \int_X f\,d\mu,$ where $G_\mu$ denotes the set of generic points for $\mu$. For factor maps $\pi$ with uniformly zero fiber-entropy, pressure is exactly preserved on images of generic sets: $$P(\pi(G_\mu),\{F_n\},f) = P(G_\mu,\{F_n\},f\circ\pi) = h_\mu(X,G) + \int f\circ\pi\,d\mu.$$ This rigidity applies to principal extensions, finite-to-one factors in higher-rank amenable groups, and classical subshift block codes.

5. Relative Uniformly Positive Entropy and Induced Systems

Relative uniformly positive entropy (rel-u.p.e.) for factor maps $T:(X,G)\to(Y,G)$ quantifies the degree of independence or chaotic behavior retained across fibers. For any two-element open cover non-dense on a fiber, the conditional entropy remains strictly positive (Liu et al., 2022): $h_{\mathrm{top}}(\{U_1,U_2\}\,|\,T)>0.$

A factor map induces a corresponding map $$\widetilde{\pi}:(\mathcal{M}(X),G)\to(\mathcal{M}(Y),G)$$ on spaces of Borel probability measures with the weak$^*$ topology. Openness of $\pi$ is preserved under induction, and rel-u.p.e. passes between $(X,G)$ and $(\mathcal{M}(X),G)$ when the factor system $Y$ is fully supported.

6. Higher-Dimensional Weighted Pressure and Factor Maps

Factor chains $\pi_i:(X_i,G)\to(X_{i+1},G)$ allow construction of weighted topological pressure $P^{\mathbf{a}}(f,G)$ incorporating weights at each level. The variational principle (Yin et al., 2023) generalizes: $$P^{\mathbf{a}}(f,G) = \sup_{\mu\in \mathcal{M}^G(X_1)}\left( \sum_{i=1}^r w_i h_{\mu_i}(X_i,G) + w_1 \int_{X_1} f\,d\mu \right),$$ with induced measures $\mu_i := \pi_{i-1} \circ \cdots \circ \pi_1 \mu$ and weights $w_i$ derived from layer parameters $a_i$.

7. Applications and Examples

• Subshifts and block codes: Factor maps arising from block code projections between subshifts maintain packing pressure, due to finite fibers ($h_{\mathrm{top}}=0$).
• Algebraic and principal extensions: In compact group actions by automorphisms, principal extensions exhibit zero fiber-entropy, ensuring pressure invariance.
• Higher-rank group extensions: For $G=\mathbb{Z}^d$, group extensions with finite fibers admit complete pressure rigidity under factor maps.
• Induced measure actions: Properties such as uniformly positive entropy and openness of factor maps are preserved from dynamical systems to their induced measure spaces (Liu et al., 2022).
• Orbit equivalence and multiorder factors: Any free, measure-preserving $G$-action admits a multiorder factor and a successor map generating a $\mathbb{Z}$-action with preservation of conditional entropy and Pinsker σ-algebra under orbit equivalence (Downarowicz et al., 2021).
• Minimal models: Every free amenable group action on a compact metric zero-dimensional space is measurably isomorphic to a minimal $G$-action with matched simplex of invariant measures (Frej et al., 2014).

In summary, factor maps for amenable group actions serve as robust tools for transferring dynamical structure, equidistribution, entropy, and pressure invariants between systems. Packing topological pressure is highly stable under factor maps, with complete invariance when fibers are topologically trivial or under strong ergodic or specification hypotheses. Extensions to higher-rank and induced measure systems exhibit preservation of core dynamical features, ensuring the utility of factor maps throughout amenable group dynamics.