Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

On the fractional susceptibility function of piecewise expanding maps

Aspenberg, Magnus LU ; Baladi, Viviane LU ; Leppänen, Juho and Persson, Tomas LU orcid (2022) In Discrete and Continuous Dynamical Systems- Series A 42(2). p.679-708
Abstract

We associate to a perturbation (ft) of a (stably mixing) piecewise expanding unimodal map f0 a two-variable fractional susceptibility function Ψφ(η, z), depending also on a bounded observable φ. For fixed η ∈ (0, 1), we show that the function Ψφ(η, z) is holomorphic in a disc Dη ⊂ C centered at zero of radius > 1, and that Ψφ(η, 1) is the Marchaud fractional derivative of order η of the function t 7→ Rφ(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 Ψφ(η, z) admits a holomorphic extension to the domain {(η, z) ∈ C2 | 0 < <η < 1, z ∈ Dη }. Finally, if the perturbation (ft)... (More)

We associate to a perturbation (ft) of a (stably mixing) piecewise expanding unimodal map f0 a two-variable fractional susceptibility function Ψφ(η, z), depending also on a bounded observable φ. For fixed η ∈ (0, 1), we show that the function Ψφ(η, z) is holomorphic in a disc Dη ⊂ C centered at zero of radius > 1, and that Ψφ(η, 1) is the Marchaud fractional derivative of order η of the function t 7→ Rφ(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 Ψφ(η, z) admits a holomorphic extension to the domain {(η, z) ∈ C2 | 0 < <η < 1, z ∈ Dη }. Finally, if the perturbation (ft) is horizontal, we prove that limη∈(0,1),η→1 Ψφ(η, 1) = ∂tRφ(t)|t=0.

(Less)
Please use this url to cite or link to this publication:
author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Fractional integrals, Fractional response, Linear response, Sobolev spaces, Transfer operators
in
Discrete and Continuous Dynamical Systems- Series A
volume
42
issue
2
pages
30 pages
publisher
American Institute of Mathematical Sciences
external identifiers
  • scopus:85123539252
ISSN
1078-0947
DOI
10.3934/DCDS.2021133
language
English
LU publication?
yes
id
87bd5a35-88e7-4ccd-88e1-2590b5a60cce
date added to LUP
2022-04-06 15:50:47
date last changed
2023-02-28 15:44:58
@article{87bd5a35-88e7-4ccd-88e1-2590b5a60cce,
  abstract     = {{<p>We associate to a perturbation (ft) of a (stably mixing) piecewise expanding unimodal map f0 a two-variable fractional susceptibility function Ψ<sub>φ</sub>(η, z), depending also on a bounded observable φ. For fixed η ∈ (0, 1), we show that the function Ψ<sub>φ</sub>(η, z) is holomorphic in a disc Dη ⊂ C centered at zero of radius &gt; 1, and that Ψ<sub>φ</sub>(η, 1) is the Marchaud fractional derivative of order η of the function t 7→ R<sub>φ</sub>(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<sub>φ</sub>(η, z) admits a holomorphic extension to the domain {(η, z) ∈ C<sup>2</sup> | 0 &lt; &lt;η &lt; 1, z ∈ Dη }. Finally, if the perturbation (ft) is horizontal, we prove that lim<sub>η</sub>∈<sub>(0,1)</sub>,η→<sub>1</sub> Ψ<sub>φ</sub>(η, 1) = ∂tR<sub>φ</sub>(t)|<sub>t</sub>=0.</p>}},
  author       = {{Aspenberg, Magnus and Baladi, Viviane and Leppänen, Juho and Persson, Tomas}},
  issn         = {{1078-0947}},
  keywords     = {{Fractional integrals; Fractional response; Linear response; Sobolev spaces; Transfer operators}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{679--708}},
  publisher    = {{American Institute of Mathematical Sciences}},
  series       = {{Discrete and Continuous Dynamical Systems- Series A}},
  title        = {{On the fractional susceptibility function of piecewise expanding maps}},
  url          = {{http://dx.doi.org/10.3934/DCDS.2021133}},
  doi          = {{10.3934/DCDS.2021133}},
  volume       = {{42}},
  year         = {{2022}},
}