Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear Response for Intermittent Circle Maps

Odaudu Etubi

TL;DR

This work develops a linear response theory for the intermittent circle map by leveraging Baladi–Todd cone techniques to obtain weak differentiability of the SRB measure with respect to parameter changes. Central to the approach is the construction of invariant cones, a random perturbation framework with distortion control, and careful transfer-operator calculus to derive a well-defined derivative formula involving the resolvent $(I-\mathcal L_\alpha)^{-1}$. The authors establish local Lipschitz continuity of the parameter-to-observable map, prove summability of the perturbative terms, and extend results to observables in $L^q$. As an application, they lift regularity to the base dynamics of a solenoid map with intermittency, proving statistical stability of the associated SRB measure under perturbations. The results combine precise control of neutral fixed-point behavior with perturbative spectral analysis, contributing to the understanding of linear response in slowly mixing systems and their skew-product extensions.

Abstract

Using the Cone technique of Baladi and Todd, we show some form of weak differentiability of the SRB measure for the intermittent circle maps, demonstrating linear response in the process. Subsequently, as an application, we lift the regularity from the base dynamics of the solenoid map with intermittency, showing that this family is statistically stable.

Linear Response for Intermittent Circle Maps

TL;DR

This work develops a linear response theory for the intermittent circle map by leveraging Baladi–Todd cone techniques to obtain weak differentiability of the SRB measure with respect to parameter changes. Central to the approach is the construction of invariant cones, a random perturbation framework with distortion control, and careful transfer-operator calculus to derive a well-defined derivative formula involving the resolvent . The authors establish local Lipschitz continuity of the parameter-to-observable map, prove summability of the perturbative terms, and extend results to observables in . As an application, they lift regularity to the base dynamics of a solenoid map with intermittency, proving statistical stability of the associated SRB measure under perturbations. The results combine precise control of neutral fixed-point behavior with perturbative spectral analysis, contributing to the understanding of linear response in slowly mixing systems and their skew-product extensions.

Abstract

Using the Cone technique of Baladi and Todd, we show some form of weak differentiability of the SRB measure for the intermittent circle maps, demonstrating linear response in the process. Subsequently, as an application, we lift the regularity from the base dynamics of the solenoid map with intermittency, showing that this family is statistically stable.
Paper Structure (16 sections, 18 theorems, 187 equations)

This paper contains 16 sections, 18 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. The intermittent circle map
  3. Asymptotic behaviour near the fixed point
  4. The mechanism
  5. Invariant cones
  6. A random perturbed operator, distortion property and decay estimate
  7. Important estimates
  8. Some properties of the transfer operator
  9. The linear response formula
  10. Well-defined formula
  11. Local Lipschitz continuity
  12. Summability of $A_n$ :
  13. Summability of $B_n$ :
  14. Convergence to the limit
  15. Observables in $L^q$
  16. ...and 1 more sections

Key Result

Theorem A

Suppose that $f_\alpha$ is the family of circle maps described above for $\alpha \in (0, 1)$ and satisfy the assumptions A1-A3. Then for any $\psi \in L^q(m)$ with $q > (1-\alpha)^{-1}$, Taking limit $\varepsilon \to 0^+$, eq:linearresponse holds for $\alpha=0$.

Theorems & Definitions (37)

  • Theorem A
  • Definition 1: Cone
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • ...and 27 more