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Linear response for intermittent maps with critical point
Dynamical Systems · Submitted 4 Jun 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables $\phi: [0,1] \to \mathbb{R}$ the bivariate map $(\alpha, \beta) \mapsto \int_0^1 \phi \, d\mu_{\alpha,\beta}$, where $\mu_{\alpha, \beta}$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.
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The abstract, sentence by sentence
We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ 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 $\phi: [0,1] \to \mathbb{R}$ the bivariate map $(\alpha, \beta) \mapsto \int_0^1 \phi \, d\mu_{\alpha,\beta}$, where $\mu_{\alpha, \beta}$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ with a neutral fixed point and a non-flat critical point. | background 0.89 | 0.36 | 0.03 | asserted 0.91 |
| s1 | Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables $\phi: [0,1] \to \mathbb{R}$ the bivariate map $(\alpha, \beta) \mapsto \int_0^1 \phi \, d\mu_{\alpha,\beta}$, where $\mu_{\alpha, \beta}$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative. | contribution 0.56 | 0.96 | 0.90 | asserted 1.00 |
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The paper
Original abstract
We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ 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 $\phi: [0,1] \to \mathbb{R}$ the bivariate map $(\alpha, \beta) \mapsto \int_0^1 \phi \, d\mu_{\alpha,\beta}$, where $\mu_{\alpha, \beta}$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.
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Cites as background 4 papers
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In this case, robustness of statistical properties have been investigated for both deterministic [ref] and random perturbations [ref] .
Rigorous Computation of Linear Response for Intermittent Maps · Introduction · 0.88
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Statistical properties of piecewise convex interval maps admitting both indifferent fixed points and critical points have been previously investigated in [ref] .
Doubly Intermittent Full Branch Maps with Critical Points and Singularities · Introduction · 1.00
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Statistical properties of piecewise convex interval maps admitting both indifferent fixed points and critical points have been previously investigated in [ref] .
Invariant measures for random piecewise convex maps · Introduction · 1.00
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