Entropy at Infinity: Asymptotic Analysis

• Entropy at infinity is a collection of concepts defining asymptotic escape rates and cumulative complexity beyond compact regions in various mathematical systems.
• It quantifies the exponential growth of escaping orbits, controls equilibrium measures, and delineates the dichotomy between compact-type and noncompact dynamics.
• Applications span dynamical systems, geometric flows, and quantum gravity, aiding in the analysis of singularity formation, nonequilibrium entropy, and black hole microstate counts.

Entropy at infinity encompasses a set of rigorous mathematical concepts that quantify asymptotic information growth, dynamical complexity, or disorder “at large scales” or “in extreme regimes” across probability theory, ergodic theory, geometry, PDE, and mathematical physics. The term refers not to a single invariant but a constellation of related notions, each capturing the limiting behavior of entropy, complexity, or associated measures in infinite or noncompact settings, or in asymptotic sequences where localized or bulk entropies remain finite but cumulative or “escape rates” diverge.

1. Entropy at Infinity in Dynamical Systems and Negative Curvature

In dynamical systems, particularly the geodesic flow on noncompact, pinched negatively curved manifolds, entropy at infinity quantifies the asymptotic exponential growth of “escaping” orbits, i.e., those spending nontrivial portions of time outside any compact subset. Multiple, ultimately equivalent, definitions are prominent:

• Topological entropy at infinity $h_\infty(g)$ is defined via the rate of exponential orbit growth outside increasing compacts. For $(M,g)$ negatively curved,

$$h_\infty(g) = \inf_{W \Subset M} \limsup_{R \to \infty} \frac{1}{R} \log \#\{\gamma \in \Gamma_W : d_g(o,\gamma o) \leq R\}$$

where $\Gamma_W$ is the set of $\pi_1(M)$-elements with long geodesic segments outside $W$ (Schapira et al., 2018).

• Measure-theoretic entropy at infinity is defined as

$$h^\infty_\mu(g) = \sup_{(\mu_n) \subset \mathcal{M}(g),\, \mu_n \to 0} \limsup_{n \to \infty} h_{\mu_n}(g)$$

with the supremum over invariant measures “escaping to infinity” (Velozo, 2017).

• Gurevič and Poincaré pressures at infinity quantify escape rates for weighted periodic orbit sums and Poincaré series restricted to excursions outside compacts (Gouëzel et al., 2020).

The fundamental result is the equivalence of these distinct notions in negative curvature:

$$h_\infty(g) = h^\infty_{\mathrm{top}}(g) = h^\infty_\mu(g) = P_\infty^{\mathrm{per}}(F) = P_\infty^{\mathrm{Poinc}}(F) = P_\infty^{\mathrm{var}}(F)$$

where $F$ is a potential in the thermodynamic formalism (Velozo, 2017, Gouëzel et al., 2020).

2. Critical Gap and Strong Positive Recurrence

A major conceptual advance enabled by these entropy-at-infinity invariants is the precise dichotomy between “compact-type” and genuinely noncompact dynamics via the critical gap condition:

$h_\infty(g) < h_{\mathrm{top}}(g)$

A manifold $(M,g)$ satisfying this is strongly positively recurrent (SPR). SPR manifolds enjoy remarkable properties: unique finite measure of maximal entropy (Bowen–Margulis), a discrete spectrum for the Laplacian, and $C^1$-regularity of entropy under metric variation—generalizing compact Anosov theory to the noncompact setting (Schapira et al., 2018). From the perspective of thermodynamic formalism, SPR corresponds to potentials $F$ with $P_\infty(F) < P(F)$, implying existence and uniqueness of finite equilibrium (Gibbs) measures (Gouëzel et al., 2020, Velozo, 2017).

Standard classes of SPR manifolds include all compact and convex-cocompact cases, geometrically finite manifolds with “cusp” critical gap, Schottky-type surface amalgams, and many variable curvature examples (Schapira et al., 2018).

3. Entropy at Infinity in Probability, Ergodic Theory, and Statistical Mechanics

In infinite or noncompact measure spaces, naive entropy computations often diverge, highlighting subtle dependence on the decay of probabilities and the support cardinality:

• Infinite Shannon entropy: For countable probability distributions $P = \{p_i\}$ with $\sum p_i = 1$, the discrete entropy $H(P) = -\sum_i p_i \ln p_i$ can diverge not due to improper normalization, but through an “infinitesimal” spread over infinitely many states (Baccetti et al., 2012). Sufficient conditions for divergence involve sequences of $p_i \to 0$ with $\sum p_i(-\ln p_i)$ unbounded and at least exponential growth in the number of nonzero-probability states. Dirichlet series (zeta function) techniques provide necessary and sufficient criteria:

$$H(P) = -\left.\frac{d}{dz} \ln \zeta_S(z)\right|_{z=1}$$

The phenomenon captures how entropy “at infinity” can appear in both quantum and statistical field theory settings, where a countably infinite (or continuum) set of modes leads to natural divergences even when modes are individually normalizable.

• Slow entropy and Kushnirenko invariants: In ergodic theory, generic measure-preserving systems exhibit “entropy at infinity” as unbounded slow entropy along sparse time progressions, despite classical entropy being zero (Ryzhikov, 2020). This captures hidden complexity visible only on certain scales, with the generic transformation displaying infinite scaled entropy (comeager subset) and a meager set of zero-entropy systems such as interval exchange transformations.
• Nonequilibrium entropy production: In Markovian or stochastic models, irreversible transitions where backward rates vanish yield infinite environmental entropy production per the Schnakenberg formula. Physically, no real system achieves exactly zero backward rates, so observed entropy production grows with the observation time as $\sim \ln T$—an “entropy at infinity” effect that encodes how rare event statistics regularize model-theoretic divergences (Zeraati et al., 2012).

4. Entropy at Infinity in Parabolic and Geometric Flows

Entropy at infinity appears as a tool for understanding singularity formation and asymptotics in geometric evolution equations:

• Curve Shortening and Mean Curvature Flow: The Colding–Minicozzi entropy measures the Gaussian-weighted local area, and its “entropy at infinity” is formalized via backward limits of ancient flows under parabolic rescaling. For ancient solutions to curve shortening flow in the plane with finite entropy, the tangent flow at infinity is unique—a possibly multiply covered line—with exponential convergence (Choi et al., 17 May 2024). For ancient mean curvature flows, Ecker's local integral at infinity and Huisken's global entropy at infinity agree, establishing that localization in space and time yield the same monotonic limit; this is essential in classifying ancient solutions and constructing blow-down limits (Kunikawa, 2020).
• Entropy-bounded solutions to compressible MHD and Navier-Stokes: In analysis of 3D compressible, heat-conducting MHD equations with vacuum at infinity, “entropy at infinity” refers to proving both upper and lower bounds (uniform in space, including as $|x| \to \infty$) for the specific entropy, despite degeneracy as density and temperature vanish. The boundedness is obtained via singular weighted energy estimates and De Giorgi iterations, indicating propagation of regularity and control of entropy at spatial infinity (Liu et al., 2022).

5. Black Hole Entropy at Infinity and Quantum Gravity

In gravitational physics and black hole thermodynamics, entropy at infinity refers both to the computation of the Bekenstein–Hawking entropy from asymptotic data and to the encoding of black hole microstate counts in boundary observables at null infinity ($\mathscr{I}^+$):

• Canonical and Holographic Entropy Computations: For asymptotically flat black holes, a dimensional reduction to 2D dilaton gravity plus a Weyl transformation maps the problem to an AdS$_2$ geometry. The black hole entropy can then be computed at infinity as the entanglement entropy of the asymptotic AdS$_2$ region, perfectly matching the conventional area law (Edholm et al., 2018). Similarly, a microstate counting via canonical quantization of the BMS$_3$ asymptotic symmetry algebra at future null infinity, coupled to 2D bulk degrees of freedom representing soft gravitons, yields a Cardy-type formula reproducing $S_{\rm BH} = A_{\rm hor}/(4G_N)$ (Shajiee, 5 Mar 2025).
• Null Infinity and BMS Charges: On cuts of future null infinity, the “renormalized area” (infinite but made finite by vacuum subtraction) shifts anomalously under supertranslations, and is linked via a conjectured bound to the outgoing entanglement entropy across the cut:

$$\frac{A_0(\Sigma)}{4\hbar G} \geq S_{\rm out}(\Sigma)$$

This elucidates the interplay between geometry, soft quantum hair, BMS symmetry, and information, positioning entropy at infinity as a primary observable for quantum gravity in asymptotically flat spacetimes (Kapec et al., 2016).

6. Overview: Unifying Features and Significance

Across these domains, entropy at infinity functions as a diagnostic for excess complexity, information, or disorder unaccounted for by classical, local, or compactly supported invariants. In all settings:

• It encodes the “escape rate” of entropy as orbits, configurations, or mass escape from any compact region, or as probability is diluted over infinite state spaces.
• Finite entropy at infinity frequently underpins strong regularity, existence, or finiteness results; infinite entropy at infinity usually signals qualitative changes in dynamical, analytic, or physical behavior.
• Rigorous criteria (growth rates, critical exponents, pressure gaps, sharp inequalities) allow precise determination of when entropy at infinity is finite, and when it is not—often dictating whether dynamical, statistical, or geometric structure resembles the compact case or not.

Entropy at infinity is thus a fundamental tool for the quantitative analysis of asymptotic phenomena in modern mathematics and theoretical physics, underpinning advances in dynamical systems, geometric analysis, statistical mechanics, and gravitational theory (Baccetti et al., 2012, Velozo, 2017, Schapira et al., 2018, Gouëzel et al., 2020, Kapec et al., 2016, Edholm et al., 2018, Shajiee, 5 Mar 2025, Kunikawa, 2020, Choi et al., 17 May 2024, Liu et al., 2022).