Limit Theorems and Quantitative Statistical Stability for the Equilibrium States of Piecewise Partially Hyperbolic Maps

This paper establishes limit theorems and quantitative statistical stability for a class of piecewise partially hyperbolic maps that are not necessarily continuous nor locally invertible. By employing a flexible functional-analytic framework that bypasses the classical requirement of compact embeddings between Banach spaces, we obtain explicit rates of convergence for the variation of equilibrium states under perturbations. Furthermore, we prove the exponential decay of correlations and the Central Limit Theorem for Hölder observables. A key feature of our approach is its applicability to systems where traditional spectral gap techniques fail due to the presence of singularities and the lack of invertibility. We provide several examples illustrating the scope of our results, including partially hyperbolic attractors over horseshoes, non-invertible dynamics semi-conjugated to Manneville–Pomeau maps, and fat solenoidal attractors.

💡 Research Summary

The paper addresses the statistical properties of a broad class of piecewise partially hyperbolic maps that may be discontinuous and non‑invertible. Classical approaches to statistical stability—most notably the Keller‑Liverani perturbation theory—rely on a pair of Banach spaces (a strong space B_s and a weak space B_w) with a compact embedding B_s ↪ B_w. This compactness is essential for invoking the Hennion‑Ionescu‑Tulcea theorem and obtaining a quasi‑compact transfer operator with a spectral gap. However, in many realistic systems (e.g., those with singularities, discontinuity curves, or non‑invertible branches) constructing such embeddings is extremely difficult or impossible.

The authors develop a flexible functional‑analytic framework that discards the compact‑embedding requirement. They work with a pair of norms (‖·‖_s, ‖·‖_w) defined on a linear space that need not be a Banach space. The transfer operator L_φ associated with a Hölder potential φ satisfies a Lasota‑Yorke type inequality in the strong norm, ‖L_φ^n u‖_s ≤ B β^n‖u‖_s + C‖u‖_w (0 < β < 1), and a uniform contraction in the complement of the leading eigen‑projector, ‖N^n u‖_s ≤ D r^n‖u‖_s (0 < r < 1). Consequently L_φ can be decomposed as L_φ = P + N with P a rank‑one projector onto the eigenfunction h (associated to the maximal eigenvalue λ > 0) and N nilpotent in the sense N P = P N = 0. This yields a “spectral gap” without any compactness assumption.

From this spectral structure the authors derive two major statistical results. First, exponential decay of correlations holds for all Hölder observables ψ, ϕ: |∫ ψ∘F^n ϕ dμ − ∫ψ dμ ∫ϕ dμ| ≤ C α^n, where α is determined by β and r. The decay is uniform despite the presence of discontinuities and the lack of invertibility. Second, using Nagaev‑Guivarc’h techniques adapted to the non‑compact setting, they prove a Central Limit Theorem: for any Hölder observable g, (S_n(g) − n∫g dμ)/√n → 𝒩(0,σ²), with σ² > 0 given by the usual Green‑Kubo formula. Thus the Birkhoff sums exhibit normal fluctuations even in these irregular dynamics.

The paper also tackles quantitative statistical stability. Consider a family of perturbed maps F_δ (δ > 0) and the associated transfer operators L_δ. The authors show that the weak‑to‑strong norm of the perturbation satisfies ‖L_δ − L_0‖_{w→s} ≤ C δ, and by extending Keller‑Liverani’s perturbation theorem they obtain an explicit modulus of continuity for the equilibrium states: d(μ_δ, μ_0) ≤ D_R(δ) · ζ · |log δ|, where ζ∈(0,1] is the Hölder exponent of the observables and D_R(δ) is a polynomial‑type function of δ. This provides a concrete rate at which the invariant measure varies with the perturbation, a significant improvement over the usual qualitative continuity results.

The dynamical setting is described in detail. The base map f : M→M satisfies a mixture of expanding and contracting behavior: on a large open set A it is uniformly expanding (expansion factor σ > 1) while on its complement it may contract, but the number of expanding branches exceeds the degree of f, ensuring enough expansion globally. The potential φ belongs to a small Hölder cone P_M, guaranteeing that the associated transfer operator fulfills the Lasota‑Yorke bounds. The fiber map G : M×K→K is uniformly contracting on almost every vertical fiber γ_x={x}×K (contraction factor α < 1) and satisfies a Hölder regularity condition (H2) that allows discontinuities only along the boundaries of the partition of M.

Several illustrative examples are provided. (i) The Manneville–Pomeau map with potentials φ_t = −t log|Df| shows how the framework handles non‑uniformly expanding maps with indifferent fixed points. (ii) A skew‑product on the unit square, F(x,y) = (3x mod 1, g(x,y)), where g is C²‑close to the identity in the fiber direction, demonstrates the treatment of non‑invertible, partially hyperbolic systems with a small amount of fiber dependence. (iii) A higher‑dimensional “fat solenoidal attractor” illustrates the applicability to systems with a dense set of singularities and a non‑trivial attractor geometry. In each case the authors verify the hypotheses (f1)–(f3) for the base and (H1)–(H2) for the fibers, then apply the abstract results to obtain exponential mixing, CLT, and quantitative stability.

In summary, the paper makes three substantial contributions: (1) it introduces a novel functional‑analytic setting that bypasses the need for compact embeddings, enabling spectral gap arguments for discontinuous, non‑invertible maps; (2) it establishes exponential decay of correlations, a Central Limit Theorem, and an explicit quantitative stability estimate for equilibrium states; (3) it demonstrates the breadth of the theory through a variety of concrete examples, including maps with indifferent fixed points, skew‑products with weak fiber dependence, and solenoidal attractors. These results significantly extend the reach of thermodynamic formalism and statistical mechanics to a class of dynamical systems that were previously inaccessible to standard spectral‑gap techniques.