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Linear response for intermittent maps with critical point

Juho Leppänen

Abstract

We consider a two-parameter family of maps $T_{α, β}: [0,1] \to [0,1]$ with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables $φ: [0,1] \to \mathbb{R}$ the bivariate map $(α, β) \mapsto \int_0^1 φ\, dμ_{α,β}$, where $μ_{α, β}$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.

Linear response for intermittent maps with critical point

Abstract

We consider a two-parameter family of maps with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of observables the bivariate map , where denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.
Paper Structure (18 sections, 22 theorems, 227 equations, 1 figure)

This paper contains 18 sections, 22 theorems, 227 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Loss of memory and distortion bounds
  4. Loss of memory
  5. Technical estimates
  6. About cones and invariant measures
  7. Regularity properties of ${\mathcal{L}}_{\gamma}(\varphi)$
  8. Proof of Theorem \ref{['thm:main']} for $\phi \in L^\infty$
  9. Limit of $R_k^{(i)}$
  10. Limit of $Q_k^{(i)}$
  11. $i=1$
  12. $i=2$
  13. Proof of Lemma \ref{['lem:cone_in_c3']}
  14. Proof of Theorem \ref{['thm:main']} for $\phi \in L^q$
  15. Loss of memory in $L^p$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\gamma = (\alpha, \beta) \in \mathfrak{D}.$ Suppose that $\phi : [0,1] \to {\mathbb R}$ is a measurable function such that holds for some $q > \tfrac{1 }{ 1 - \alpha \beta}$. Then, for $i \in \{1,2\}$, is absolutely summable. Moreover, ${\mathcal{R}}_\phi$ is differentiable at $\gamma$ with directional derivative for any unit vector $v = (v_1, v_2) \in {\mathbb R}^2$.

Figures (1)

  • Figure 1: Intermittent behavior of trajectories $T_{\alpha,\beta}^n(x)$ for $(\alpha, \beta) = (0.5, 1.5)$ and $(\alpha, \beta) = (0.5, 1.0)$. The latter case corresponds to the standard Liverani--Saussol--Vaienti map with parameter $\alpha = 0.5$. Once the trajectory lands near $\tfrac{1}{2}$ in $(\tfrac{1}{2}, 1]$, after one iterate it will be in a small neighborhood of $0$ and the larger the $\alpha\beta$ the longer it will take for the trajectory to return to the strongly chaotic region.

Theorems & Definitions (51)

  • Theorem 1.1: Linear response
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 41 more