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Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Stefano Galatolo, Rafael Lucena

TL;DR

The paper develops a spectral-gap framework for transfer operators of systems preserving a contracting foliation, using anisotropic spaces built from leafwise disintegrations. By proving Lasota-Yorke inequalities and a Perron-Frobenius–type decomposition, it obtains exponential convergence to equilibrium and a spectral gap for the transfer operator on both L^1-like and L^∞-like strong spaces. The framework is then applied to Lorenz-like two-dimensional maps to derive quantitative statistical stability under deterministic perturbations, with an asymptotically optimal modulus of continuity O(δ log δ) in BV-like norms. A uniform BV Lorenz-like family is studied to provide robust stability results and explicit rates, supported by extensive Appendix technicalities on disintegration and operator uniformity. This work advances quantitative understanding of statistical stability for piecewise smooth, fiber-contracting dynamics, with practical implications for Lorenz-like systems and related skew-product models.

Abstract

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $δ$, the physical measure varies continuously, with a modulus of continuity $O(δ\log δ)$, which is asymptotically optimal for this kind of piecewise smooth maps.

Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

TL;DR

The paper develops a spectral-gap framework for transfer operators of systems preserving a contracting foliation, using anisotropic spaces built from leafwise disintegrations. By proving Lasota-Yorke inequalities and a Perron-Frobenius–type decomposition, it obtains exponential convergence to equilibrium and a spectral gap for the transfer operator on both L^1-like and L^∞-like strong spaces. The framework is then applied to Lorenz-like two-dimensional maps to derive quantitative statistical stability under deterministic perturbations, with an asymptotically optimal modulus of continuity O(δ log δ) in BV-like norms. A uniform BV Lorenz-like family is studied to provide robust stability results and explicit rates, supported by extensive Appendix technicalities on disintegration and operator uniformity. This work advances quantitative understanding of statistical stability for piecewise smooth, fiber-contracting dynamics, with practical implications for Lorenz-like systems and related skew-product models.

Abstract

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size , the physical measure varies continuously, with a modulus of continuity , which is asymptotically optimal for this kind of piecewise smooth maps.

Paper Structure

This paper contains 26 sections, 61 theorems, 248 equations.

Table of Contents

  1. Contracting Fiber Maps
  2. Properties of $G$
  3. Properties of $T$ and of its associated transfer operator.
  4. Weak and strong spaces
  5. $L^{1}$-like spaces.
  6. The $\mathcal{L}^{1}$ and $S^1$ spaces
  7. $L^{\infty }$ like spaces
  8. The transfer operator associated to $F$
  9. Basic properties of the norms and convergence to equilibrium
  10. Convergence to equilibrium
  11. $L^{\infty }$ norms
  12. Spectral gap
  13. Exponential Decay of Correlations
  14. Application to Lorenz-like maps
  15. Properties of $T_{L}$ in Lorenz-like systems
  16. ...and 11 more sections

Key Result

Theorem 1.1

If $T$ satisfy $T3.1,...,T3.4$ then there exist $0<r<1$ and $D>0$ such that for all $f\in S_{\_}$ with $\int {f~}dm_{1}=0$ and for all $n\geq 0,$ it holds

Theorems & Definitions (131)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Lemma 2.2
  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Proposition 2.3
  • Definition 4
  • ...and 121 more