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Linear response for systems with a cusp

Davrbek Oltiboev

TL;DR

This work addresses stability of invariant measures for dynamical systems with cusp singularities by proving a uniform spectral gap for the transfer operators acting on Sobolev spaces $W^{1,p}$ and $W^{2,p}$ (for $p>1$) and deriving an explicit $L^p$ linear response formula. The authors show that the invariant densities depend differentiably on the perturbation parameter, with $h_\varepsilon = h_0 + \varepsilon (I-P_{T_0})^{-1} q + o(\varepsilon)$ and an explicit cusp-driven kernel $q$, obtained via a perturbative expansion and the Keller–Liverani stability theorem. They also construct a concrete family of cusp maps satisfying assumptions (A1)-(A9) to illustrate the applicability of the theory and verify regularity and perturbation properties. Overall, the paper extends linear response results to systems with cusp singularities, providing sharp $L^p$-type insights and broadening the set of systems where invariant measures respond smoothly to perturbations.

Abstract

In this note we consider a tent-like family with a cusp at the singular point and show that the linear response holds for certain perturbations of this family. This contrasts the tent-like maps with finite derivatives at the singularity. Our results extend the results of Bahsoun and Galatolo to the larger class of singularities and we obtain the linear response formula in $L^p$ for $p>1$.

Linear response for systems with a cusp

TL;DR

This work addresses stability of invariant measures for dynamical systems with cusp singularities by proving a uniform spectral gap for the transfer operators acting on Sobolev spaces and (for ) and deriving an explicit linear response formula. The authors show that the invariant densities depend differentiably on the perturbation parameter, with and an explicit cusp-driven kernel , obtained via a perturbative expansion and the Keller–Liverani stability theorem. They also construct a concrete family of cusp maps satisfying assumptions (A1)-(A9) to illustrate the applicability of the theory and verify regularity and perturbation properties. Overall, the paper extends linear response results to systems with cusp singularities, providing sharp -type insights and broadening the set of systems where invariant measures respond smoothly to perturbations.

Abstract

In this note we consider a tent-like family with a cusp at the singular point and show that the linear response holds for certain perturbations of this family. This contrasts the tent-like maps with finite derivatives at the singularity. Our results extend the results of Bahsoun and Galatolo to the larger class of singularities and we obtain the linear response formula in for .

Paper Structure

This paper contains 7 sections, 8 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:main']}
  3. Uniform spectral gap on $W^{1,p}$ and $W^{2,p}$
  4. Classical results.
  5. Uniform Spectral Gap for Transfer Operators
  6. Linear Response Formula
  7. A family of maps satisfying assumptions (A1)-(A9)

Key Result

Theorem 1.1

There exists $\delta_{2}>0$ such that for all $\varepsilon \in\left[0, \delta_{2}\right)$, $T_{\varepsilon}$ admits a unique invariant probability density $h_{\varepsilon} \in W^{2,p}$. Moreover, the map $\varepsilon \mapsto h_{\varepsilon}$ is differentiable in $L^{p}$ at $\varepsilon=0$. Furthermo where with and the error term $o(\varepsilon)$ is understood in the $L^{p}$-topology.

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4: Lasota-Yorke Type Inequality
  • proof
  • Lemma 2.5: Second-Order Regularity Estimate
  • proof
  • Lemma 2.6
  • proof
  • ...and 3 more