Strong Positive Recurrence

• Strong positive recurrence is a property in dynamical systems and Markov processes that ensures robust recurrence, exponential return times, and the presence of a spectral gap.
• It underpins statistical properties such as exponential mixing, central limit theorems, and rapid decay of correlations in models ranging from countable Markov shifts to geodesic flows.
• SPR provides critical criteria in various fields—including ergodic theory, Riemannian geometry, and number theory—by establishing pressure gaps and guaranteeing unique equilibrium measures.

Strong positive recurrence (SPR) is a concept with far-reaching implications across ergodic theory, thermodynamic formalism, Markov processes, symbolic dynamics, Riemannian geometry, and number theory. SPR conditions characterize systems exhibiting robust recurrence and mixing properties, exponential tails for return times, spectral gap in associated operators, uniqueness and finiteness of equilibrium states, and—depending on context—structural stability under perturbations. The notion arises naturally in the paper of countable Markov shifts, geodesic flows, symbolic representations of dynamical systems, nonnegative matrices, and even the dynamics of L-functions under complex translation.

1. Foundational Definitions Across Contexts

Markov Chains and Nonnegative Matrices

Consider an irreducible countable nonnegative matrix $A=(A(x,y))_{x,y\in S}$ with spectral radius

$p(A) = \lim_{n\to\infty}(A^n(x,x))^{1/n}.$

The matrix $A$ is strongly R-positive if the associated recurrent Markov chain (given by the normalized kernel via Perron-Frobenius theory) returns to every state with exponential moments: $\mathbb E_z[e^{\alpha \tau_z}] < \infty$ for some $\alpha > 0$ and all $z$ ($\tau_z$ is the return time to $z$). Equivalently, for every finite modification $B < A$ (i.e., $B(x,y) \leq A(x,y)$ with $B(x,y) < A(x,y)$ only for finitely many $(x,y)$), one has $p(B) < p(A)$ (Swart, 2017).

Countable Markov Shifts

Let $M = (M_{i,j})_{i,j \in A}$ be a $0$–$1$ matrix over a countable alphabet $A$ with shift space $\Sigma$ and a summable-variation potential $\varphi: \Sigma \to \mathbb R$. Defining partition sums

$$Z_n(\varphi,a) = \sum_{\substack{x: \sigma^n x = x, x_0 = a}} \exp(S_n\varphi(x)), \quad S_n\varphi(x) = \sum_{i=0}^{n-1}\varphi(\sigma^i x),$$

the Gurevich pressure is $P_G(\varphi) = \limsup_{n\to\fty} \frac{1}{n} \log Z_n(\varphi,a)$. $\varphi$ is strongly positively recurrent (SPR) if

$$\limsup_{n\to\infty} \frac{1}{n}\log Z_n^*(\varphi,a) < P_G(\varphi)$$

where $Z_n^*$ counts first-return loops to $a$ (Todd et al., 4 Mar 2024).

Geodesic Flows on Noncompact Manifolds

Let $(M,g)$ be a complete, simply-connected Riemannian manifold of pinched negative curvature, with $\Gamma$ a nonelementary discrete group so $M/\Gamma$ is noncompact. For a Hölder potential $F:T^1M \to \mathbb R$, the pressure at infinity is defined via three coinciding approaches (Gurevič, geometric—via critical exponent of a restricted Poincaré series, and variational). $F$ is SPR if

$$P_\infty(F) = \delta_\Gamma^\infty(F) < P_{\text{top}}(F) = \delta_\Gamma(F)$$

so there is a strict "pressure gap" at infinity (Gouëzel et al., 2020).

Diffeomorphisms on Compact Manifolds

For a $C^{1+}$ diffeomorphism $f:M\to M$, a Borel set $X\subset M$ is $\chi$-SPR if for all $\epsilon > 0$ there are $(\chi, \epsilon)$-Pesin blocks $\Lambda \subset X$ of positive measure (invariant measures with entropy $>h_0$ assign positive mass to $\Lambda$) (Buzzi et al., 13 Jan 2025).

Riemann Zeta Function and L-functions

For a compact $K\subset\{s\in\mathbb C : 1/2< \Re(s)<1\}$, $\zeta(s)$ has strong recurrence (positive lower density) if for all $\varepsilon>0$,

$$\liminf_{T\to\infty} \frac{1}{T} \operatorname{meas}\{\tau\in[0,T]: \sup_{s\in K} |\zeta(s+i\tau)-\zeta(s)|<\varepsilon \} > 0$$

and generalized strong recurrence for parameter $d$ if $\zeta(s+i\tau)$ and $\zeta(s+id\tau)$ come close in this sense (Pańkowski, 2015).

2. Equivalent Characterizations and Theoretical Criteria

Table: Characterizations of SPR by Setting

Domain	Characterization	Reference
Nonneg. Matrix	Spectral radius drops under any finite perturbation	(Swart, 2017)
Markov Shift	Lim sup of first return partition sum exponential growth < top pressure; spectral gap exists	(Todd et al., 4 Mar 2024)
Geodesic Flow	Pressure at infinity < topological pressure	(Gouëzel et al., 2020)
Diffeomorphisms	High-entropy measures see uniform Pesin blocks of positive measure; Markov shift coding is SPR	(Buzzi et al., 13 Jan 2025)
$\zeta$-function	Approximation by jointly shifted copies occurs for positive lower density of shifts	(Pańkowski, 2015)

In Markov-type settings, SPR is equivalent to existence of a spectral gap for the transfer (Ruelle) operator, which in turn yields exponential decay of correlations and statistical limit theorems. For matrices, strong R-positivity is equivalent to exponential moments for return times and to spectral robustness, as above. In geometric dynamics, the pressure gap at infinity ensures recurrence of orbits into compact sets and finiteness of invariant Gibbs measures.

3. Constructions, Criteria, and Sharpness

Explicit criteria and constructions of SPR potentials or systems:

• Markov shifts: If a system is topologically transitive with mild boundary entropy growth and the potential has summable variations, then precise contraction at infinity ensures SPR. Failure of the strict contraction can yield positive recurrence without a spectral gap (Todd et al., 4 Mar 2024).
• Manifolds/geodesic flows: Adding a large compactly supported "bump" to a potential boosts topological pressure without affecting pressure at infinity, yielding SPR for large weights. Cusp geometry or decay to constants at infinity supplies further examples (Gouëzel et al., 2020).
• Nonnegative matrices: Any irreducible nonnegative matrix becomes strongly R-positive if lowering finitely many entries drops the spectral radius. For instance, localized pinning models provide prototypical cases—at criticality, one is positive recurrent but not strongly R-positive (Swart, 2017).
• Surface diffeomorphisms: All $C^\infty$ diffeomorphisms with positive topological entropy are SPR (Buzzi et al., 13 Jan 2025).
• Riemann zeta function: SPR is established for rational and irrational parameters $d\ne 0,\pm1$, via approximation by Euler products and Kronecker’s/Weyl’s theorem, ensuring a positive density of recurrence shifts (Pańkowski, 2015).

4. Statistical and Dynamical Significance

SPR guarantees advanced ergodic and statistical properties, including:

• Exponential mixing: Unique equilibrium (Gibbs) measure exhibits exponential decay of correlations. This applies to SPR Markov shifts (Todd et al., 4 Mar 2024), flows on noncompact manifolds (Gouëzel et al., 2020), and diffeomorphisms via symbolic coding (Buzzi et al., 13 Jan 2025).
• Central limit and invariance principles: Almost Sure Invariance Principle (ASIP), CLT, law of iterated logarithm, etc., hold for global observables in the equilibrium state for SPR systems (Buzzi et al., 13 Jan 2025).
• Large deviations: Standard (Gartner–Ellis) large deviations estimates are available under SPR (Buzzi et al., 13 Jan 2025).
• Counting and renewal asymptotics: For geodesic flows, SPR ensures the asymptotic for weighted count of closed geodesics holds with sharp constants (Gouëzel et al., 2020). For Markovian systems, first return and loop statistics exhibit exponential tails.

5. Proof Strategies and Theoretical Consequences

A recurring methodology is a renewal-theoretic, spectral, or operator-theoretic approach:

• Markov shift/diffeomorphism coding: Countable Markov partitions and SPR symbolic models allow the transfer of statistical and spectral properties from shifts to the original system (Buzzi et al., 13 Jan 2025, Todd et al., 4 Mar 2024).
• Spectral perturbations: The criterion for strong R-positivity via finite modifications and analytic extension of generating functions gives a robust method for demonstrating or negating SPR in matrix settings (Swart, 2017).
• Pressure gap functional: For negative curvature, verifying a strict gap between pressure at infinity and topological pressure establishes finiteness of equilibrium measures using Patterson–Sullivan–Gibbs techniques (Gouëzel et al., 2020).

6. Illustrative Examples and Borderline Cases

• Bouquet shifts with large loop growth but logarithmic return penalties show positive recurrence without SPR (no spectral gap) (Todd et al., 4 Mar 2024).
• Pinning models at criticality demarcate the positive-but-not-strongly positive recurrent regime, providing polynomial versus exponential return time tails (Swart, 2017).
• Geometrically finite manifolds with small cusps and well-chosen potentials realize SPR conditions (Gouëzel et al., 2020).
• Non-Anosov, robustly transitive diffeomorphisms (e.g., Bonatti–Viana examples) exhibit SPR despite departing from classical uniform hyperbolicity (Buzzi et al., 13 Jan 2025).

7. SPR in Number Theory: Universality and Recurrence of L-functions

In the context of the Riemann zeta function, strong recurrence relates directly to universality phenomena and to deep conjectures in analytic number theory. For $d=0$, strong recurrence is equivalent to the Riemann Hypothesis; for $d\ne 0, \pm1$ (rational or irrational), generalized strong recurrence (joint universality of shifts of $\zeta(s)$) has been established via parameter perturbation, approximation by partial Euler products, and almost-periodicity results (Pańkowski, 2015).

This formalizes a dynamical viewpoint: translation flows in the space of holomorphic functions admit recurrent returns (in uniform norm on compacts) not only to single function values but simultaneously along distinct linear subflows.

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The concept of strong positive recurrence is thus central in describing systems—combinatorial, geometric, dynamical, probabilistic, or analytic—that combine robust recurrence with rich statistical structure and spectral regularity. Its unifying role in local-to-global phenomena, spectral-stability criteria, and stochastic laws is well documented in recent literature (Todd et al., 4 Mar 2024, Gouëzel et al., 2020, Buzzi et al., 13 Jan 2025, Swart, 2017, Pańkowski, 2015).