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Linear response due to singularities

Wael Bahsoun, Stefano Galatolo

Abstract

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a \emph{cusp} at the turning point, we recover the linear response. More precisely, let $T_\eps$ be a family of such cusp maps generated by changing the value of the turning point of $T_0$ by a deterministic perturbation and let $h_\eps$ be the corresponding invariant density. We prove that $\eps\mapsto h_\eps$ is differentiable in $L^1$ and provide a formula for its derivative.

Linear response due to singularities

Abstract

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a \emph{cusp} at the turning point, we recover the linear response. More precisely, let be a family of such cusp maps generated by changing the value of the turning point of by a deterministic perturbation and let be the corresponding invariant density. We prove that is differentiable in and provide a formula for its derivative.
Paper Structure (9 sections, 8 theorems, 71 equations, 2 figures)

This paper contains 9 sections, 8 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Family of maps and statement of the main result
  3. Proof of Theorem \ref{['thm:main']}
  4. Uniform spectral gap on $W^{1,1}$ and on $W^{2,1}$
  5. Classical results
  6. Uniform spectral gap for the associated transfer operators
  7. Linear response derivation
  8. A family of maps satisfying assumptions $(A1),...,(A9)$
  9. Concluding remarks

Key Result

Theorem 1

There is $\delta_2 >0$ such that for $\varepsilon\in[0,\delta_2 )$$T_\varepsilon$ admits a unique invariant probability density $h_\varepsilon\in W^{2,1}$; moreover, ${\varepsilon}\mapsto h_{\varepsilon}$ is differentiable in $L^1$ at $\varepsilon=0$. In particular, here with and the $o$ is in the $L^1$-topology.

Figures (2)

  • Figure 1: An example of $T_\varepsilon$ with $a_\varepsilon=0.89$ and $c=\frac{1}{2}$.
  • Figure 2: The integration domains in \ref{['scambio']}.

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • proof
  • Remark 1
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 4 more