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On the fractional susceptibility function of piecewise expanding maps

We are grateful to Daniel Smania, some of whose ideas in the collaboration [11] were very useful here. Part of this work was carried out at the Centre for Mathematical Sciences, Lund University, during VB's Knut and Alice Wallenberg Guest Professorship. VB's and JL's research is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 787304)

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  • We associate to a perturbation $ (f_t) $ of a (stably mixing) piecewise expanding unimodal map $ f_0 $ a two-variable fractional susceptibility function $ \Psi_\phi(\eta, z) $, depending also on a bounded observable $ \phi $. For fixed $ \eta \in (0,1) $, we show that the function $ \Psi_\phi(\eta, z) $ is holomorphic in a disc $ D_\eta\subset \mathbb{C} $ centered at zero of radius $ >1 $, and that $ \Psi_\phi(\eta, 1) $ is the Marchaud fractional derivative of order $ \eta $ of the function $ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $, at $ t = 0 $, where $ \mu_t $ is the unique absolutely continuous invariant probability measure of $ f_t $. In addition, we show that $ \Psi_\phi(\eta, z) $ admits a holomorphic extension to the domain $ \{\, (\eta, z) \in \mathbb{C}^2\mid 0<\Re \eta <1, \, z \in D_\eta \,\} $. Finally, if the perturbation $ (f_t) $ is horizontal, we prove that $ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $.

    Mathematics Subject Classification: Primary: 37E05; Secondary: 26A33, 37C30.


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