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Linear response asymmetry between SRB and physical measures for families of intermittent maps with a transition point

Yuya Arima

Abstract

We study linear response for families of intermittent maps whose SRB measure undergoes a transition from finite to infinite total mass at a critical parameter value. Our results reveal the following fundamental asymmetry arising from this transition. Smooth parameter dependence of the SRB measure implies continuity of the physical measure at the transition point, while simultaneously precluding its differentiability there. In particular, although the physical measure varies continuously with respect to the parameter at the transition, it fails to admit a linear response for a large class of potentials in the usual sense. We derive an explicit one-sided derivative formula describing this singular behavior and thereby give a quantitative characterization of how statistical properties degenerate as the total mass of the SRB measure diverges. The key ingredient in the proof of our main theorem is a new method that relates the parameter dependence of physical measures near the transition point to the behavior of the Riemann zeta function near its pole at 1.

Linear response asymmetry between SRB and physical measures for families of intermittent maps with a transition point

Abstract

We study linear response for families of intermittent maps whose SRB measure undergoes a transition from finite to infinite total mass at a critical parameter value. Our results reveal the following fundamental asymmetry arising from this transition. Smooth parameter dependence of the SRB measure implies continuity of the physical measure at the transition point, while simultaneously precluding its differentiability there. In particular, although the physical measure varies continuously with respect to the parameter at the transition, it fails to admit a linear response for a large class of potentials in the usual sense. We derive an explicit one-sided derivative formula describing this singular behavior and thereby give a quantitative characterization of how statistical properties degenerate as the total mass of the SRB measure diverges. The key ingredient in the proof of our main theorem is a new method that relates the parameter dependence of physical measures near the transition point to the behavior of the Riemann zeta function near its pole at 1.

Paper Structure

This paper contains 6 sections, 8 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Precise statement of the main theorem in a general framework
  3. Proofs of the results in Section \ref{['sec abstruct setting']}
  4. Examples of families of one-dimensional intermittent maps
  5. Proof of Theorem \ref{['thm main abstract']}
  6. Linear response for Pomeau–Manneville maps at the transition point

Key Result

Lemma 1.2

Let $X$ be a metric space and let $m$ be a Borel probability measure on $X$. Let $f:X\rightarrow X$ be a measurable map and let $\nu$ be a $\sigma$-finite infinite Borel measure on $X$. Suppose that $\nu$ is invariant ergodic and conservative with respect to $f$ and absolutely continuous with respec has positive measure with respect to $m$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • Example 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Lemma 4.1
  • ...and 4 more