Maximum Entropy in Hyperbolic Space

• Maximum Entropy Distribution in Hyperbolic Space is a framework that defines optimal probability measures within negatively curved geometries exhibiting exponential volume growth.
• It integrates methods from dynamical systems, information geometry, and spectral theory to derive invariant measures and canonical densities for practical applications.
• The analysis extends to applications in cosmological models, network theory, and geometric flows, providing actionable insights into high-dimensional statistical inference.

Maximum entropy distributions in hyperbolic space constitute a multifaceted topic, linking dynamical systems, information geometry, spectral theory, geometric analysis, statistical inference, and cosmological thermodynamics. The hyperbolic setting introduces negative curvature and exponential volume growth, which fundamentally modifies both the combinatorial and geometric landscape compared to Euclidean contexts. The salient forms of maximum entropy arise as invariant measures for dynamical systems, as canonical densities for natural exponential families, as optimal geometric responses for hypersurfaces undergoing flow or boundary constraints, and as limiting states in cosmological models. This entry critically organizes the core mathematical structures, variational principles, explicit models, and asymptotic characterizations of maximum entropy distributions over hyperbolic spaces.

1. Maximum Entropy: Variational Principles in Hyperbolic Dynamical Systems

In hyperbolic dynamics, the topological entropy of a system quantifies the exponential complexity of its orbits. The measure of maximal entropy, typically a Markov measure or ergodic equilibrium measure, is defined via the variational principle: $$h_{\mathrm{top}}(f) = \sup_{\mu \in \mathcal{M}_f} h(\mu)$$ where $h(\mu)$ is the Kolmogorov–Sinai (measure-theoretic) entropy, and $\mathcal{M}_f$ is the set of %%%%2%%%%-invariant probability measures for a diffeomorphism $f$ of a compact manifold. For uniformly hyperbolic dynamical systems (e.g., Anosov flows or horseshoes), the measure maximizing entropy is unique and exhibits strong statistical mixing.

The space of hyperbolic ergodic measures, $\mathcal{M}_e$, supported on an isolated homoclinic class (see (Gorodetski et al., 2015)), is both path-connected and entropy-dense under mild assumptions:

• If all hyperbolic periodic points are homoclinically related, then $\mathcal{M}_e$ is connected in the weak$^*$ metric.
• Every invariant measure in this class can be approximated (in both measure and entropy) by ergodic ones whose entropy approaches that of the original.

This topological and algebraic richness ensures that the maximum entropy distribution is not isolated, but emerges as a limit within a connected, dense family. Approximation by Markov and periodic measures facilitates numerical and theoretical calculations, and supports thermodynamic formalism for equilibrium states in hyperbolic settings.

2. Entropy, Growth, Critical Exponents, and Fractal Geometry

For groups and semigroups of isometries acting on Gromov–hyperbolic spaces, the entropy can be equated with the critical exponent—the exponential rate of orbital growth. The seminal theorem of Patterson and Sullivan ties the critical exponent, $h_r$, to the Hausdorff (visual) dimension of the limit set (see (Mercat, 2016)): $\dim_{\mathrm{vis}}(\Lambda) = h_r$ This identification holds for discrete groups and extends to convex co-compact semigroups, with the Schottky subsemigroups providing lower bounds and approximations for the critical exponent.

Entropy in this context quantifies the maximal rate at which distinct group elements can populate neighborhoods in the space, and the extremal measures (Patterson–Sullivan measures) capture the statistically “most uniformly distributed” orbits on the boundary at infinity. The lower semicontinuity of entropy under geometric limits, established in the context of Kenyon semigroups, reflects robustness of entropy-maximizing structures under perturbation and scaling, and has implications for rigidity and dimension conjectures.

3. Exponential Family Distributions: Information Geometry of Hyperbolic Models

In probabilistic and information-theoretic contexts, exponential families over hyperbolic sample spaces provide canonical maximum entropy distributions (see (Nielsen et al., 2022)). The two principal models are:

• Poincaré distributions (on the upper-half-plane $H$), parameterized by positive-definite $2\times2$ matrices or three-tuples $(a, b, c)$ with $ac - b^2 > 0$.
• Hyperboloid distributions (on the forward sheet $\mathbb{L}^d$), parameterized by $(\theta_0, \theta_1, ..., \theta_d)$ with $\theta_0 > \sqrt{\theta_1^2 + ... + \theta_d^2}$.

The log-density is linear in sufficient statistics, and the Fisher information metric and Amari–Chentsov cubic tensors reveal the dually flat structure of these families. Explicit closed-form formulas are given for Kullback–Leibler divergence, $f$-divergences, differential entropy, and Bhattacharyya distances (e.g., for Poincaré distribution

$$h[p_\theta] = 1 + \log(\pi D) - 2 \log a - 2 e^{4D} \Gamma(0,4D)$$

with $D = \sqrt{ac - b^2}$).

Mixtures of hyperboloid distributions are universal smooth density estimators for functions on hyperbolic spaces, analogous to Gaussian mixtures in the Euclidean setting. This universality is central to statistical inference and machine learning over hyperbolic geometry, enabling clustering, density estimation, and representation learning with maximum entropy priors.

4. Maximum Entropy and Optimal Hypersurfaces: Geometric Entropy in Hyperbolic Spaces

Geometric entropy for submanifolds in hyperbolic space is modeled after the Colding–Minicozzi entropy (see (Bernstein, 2020, Bernstein et al., 2023)). For an $n$-dimensional submanifold $\Sigma \subset \mathbb{H}^{n+k}$, entropy is defined by

$$\lambda_{\mathbb{H}}[\Sigma] = \sup_{p_0 \in \mathbb{H}^{n+k}, \tau > 0} \int_\Sigma \Phi_n^{(0, p_0)}(-\tau, p)\, d\mathrm{Vol}_\Sigma(p)$$

with $\Phi_n$ the hyperbolic heat kernel weight.

Key properties:

• Entropy is monotone non-increasing along mean curvature flow (MCF) for $n \leq 4$.
• Totally geodesic hyperbolic $n$-planes have minimal entropy ($\lambda_{\mathbb{H}}[\Sigma] = 1$), while proper minimal hypersurfaces possess higher entropy.
• For submanifolds with well-defined boundary at infinity, entropy is bounded below by the conformal volume of the boundary—encoding asymptotic geometry (Bernstein, 2020, Bernstein et al., 2023): $$\lambda_{\mathbb{H}}[\Sigma] \geq \frac{\lambda_c[\Gamma]}{\mathrm{Vol}(S^{n-1})}$$

Relative entropy of hypersurfaces (Yao, 2022) is defined via renormalized area differences, capturing conformally invariant energy-like quantities. Monotonicity under MCF and independence from the choice of boundary-defining function elevate relative entropy to a geometric analogue of thermodynamic entropy difference, with relevance for variational problems and AdS/CFT correspondence.

5. Asymptotics, Scaling, and Universal Canonical Measures

The unique maximum entropy measure on a compact metric space $(X, d)$, with similarity kernel $K(x, y) = e^{-d(x, y)}$, is realized by the “balanced” measure $\mu^*$ satisfying $K\mu^*(x) = c$ on its support (Leinster et al., 2019): $(K\mu^*)(x) = \int_X e^{-d(x, y)}\, d\mu^*(y) = c$ This measure maximizes a one-parameter family of generalized divergences ($q$-diversities, including Hill numbers and Rényi entropies). As the metric is rescaled ($d \mapsto t d$), the asymptotics of the maximum entropy quantify geometric invariants: $$\lim_{t \to \infty} \frac{\log\{tX\}}{\log t} = \mathrm{Dim}_{\mathrm{Mink}}(X), \qquad \lim_{t \to \infty} \frac{\{tX\}}{t^n} \propto \mathrm{Vol}(X)$$ For metric spaces with exponential volume growth (e.g., hyperbolic balls), the maximum entropy measure—or “uniform measure”—is scale-invariant and reflects the underlying geometry, supplying a canonical reference distribution even in the absence of a natural Haar measure.

6. Cosmological Entropy Maximization and Dimensional Selection

In cosmology, the principle of maximum entropy is invoked to select the ground state in de Sitter space (Volovich, 2023). For the $D$-dimensional de Sitter spacetime, the entropy is

$S(\ell, D) = \tfrac{1}{4}\,\ell^{D-2} \Omega_{D-2}$

where $\Omega_{D-2}$ depends on the Gamma function. Extremizing $S$ over $D$ (treated as continuous) yields a digamma equation whose solution selects $D=4$ and uniquely determines the inflationary cosmological constant: $$\log(\pi \ell^2) - \psi(3/2) = 0, \quad \Lambda = 3\pi\,\exp\{-\psi(3/2)\} \approx 9.087$$ This geometric entropy maximization provides both the "most probable" spacetime dimension and the relevant cosmological parameters for the early universe, illustrating how entropy extremization principles transcend mathematics into physical cosmology.

7. Extreme Value Statistics in Hyperbolic Random Graphs

Random hyperbolic graphs, fundamental to the modeling of real-world complex networks, demonstrate that degree distributions inherit maximum entropy characteristics from the underlying hyperbolic geometry (Gassmann, 9 Apr 2024):

• Node degree is strictly ordered by increasing proximity to the center, matching the ordering of distance, with maximal degrees corresponding to nodes closest to the origin.
• In the dense regime ($\alpha < 1/2$), the maximum degree converges to Weibull law; in the sparse regime ($\alpha > 1/2$), to Fréchet law; at criticality ($\alpha = 1/2$), an exponential law emerges.
• The interplay between curvature parameter ($\alpha$) and degree presentation exemplifies a phase transition, with entropy maximization manifesting as heavy-tailed distributions under geometric constraints.

These extreme value statistics underpin the emergence of “hub” structures and scale-free behavior in large-scale hyperbolic networks, linking entropy optimization with network topology.

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The mathematical and physical manifestations of maximum entropy distributions in hyperbolic space thus range from measures of maximal dynamical complexity, over information-geometric exponential families, through geometric flows and boundary invariants, to cosmological selection and network extremal statistics. The negative curvature and exponential volume growth endow these distributions with unique properties not observed in Euclidean settings, anchoring both theory and application in the geometry of hyperbolic spaces.