## Abstract

We associate to a perturbation (ft) of a (stably mixing) piecewise expanding unimodal map f0 a two-variable fractional susceptibility function Ψ_{φ}(η, z), depending also on a bounded observable φ. For fixed η ∈ (0, 1), we show that the function Ψ_{φ}(η, z) is holomorphic in a disc Dη ⊂ C centered at zero of radius > 1, and that Ψ_{φ}(η, 1) is the Marchaud fractional derivative of order η of the function t 7→ R_{φ}(t):= R φ(x) dµt, at t = 0, where µt is the unique absolutely continuous invariant probability measure of ft. In addition, we show that Ψ_{φ}(η, z) admits a holomorphic extension to the domain {(η, z) ∈ C^{2} | 0 < <η < 1, z ∈ Dη }. Finally, if the perturbation (ft) is horizontal, we prove that lim_{η}∈_{(0,1)},η→_{1} Ψ_{φ}(η, 1) = ∂tR_{φ}(t)|_{t}=0.

Original language | English |
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Pages (from-to) | 679-708 |

Number of pages | 30 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 42 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2022 Feb |

## Subject classification (UKÄ)

- Mathematical Analysis

## Free keywords

- Fractional integrals
- Fractional response
- Linear response
- Sobolev spaces
- Transfer operators