25 research outputs found
Fractional susceptibility functions for the quadratic family: Misiurewicz-Thurston parameters
For the quadratic family, we define the two-variable ( and )
fractional susceptibility function associated to a C^1 observable at a
stochastic map. We also define an approximate, "frozen" fractional
susceptibility function. If the parameter is Misiurewicz-Thurston, we show that
the frozen susceptibility function has a pole at for generic observables
if a "one-half" transversality condition holds. We introduce "Whitney"
fractional integrals and derivatives on suitable sets . We formulate
conjectures supported by our results on the frozen susceptibility function and
numerical experiments. In particular, we expect that the fractional
susceptibility function for is singular at for Collet-Eckmann
maps and generic observables. We view this work as a step towards the
resolution of the paradox that the classical susceptibility function is
holomorphic at for Misiurewicz-Thurston maps, despite lack of linear
response.Comment: Version v3 is the electronic copy of the published version in Comm
Math Phys. 4 figure
Edgeworth expansions for slow-fast systems with finite time scale separation
We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter ε. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in ε and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation
Gibbs-Markov-Young Structures and Decay of Correlations
In this work we study mixing properties of discrete dynamical systems and related to them geometric structure. In the first chapter we show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise interval maps with critical points and singularities, H\'enon maps and partially hyperbolic systems.
The second chapter is dedicated to the problem of decay of correlations for continuous observables. First we show that if the underlying system admits Young tower then the rate of decay of correlations for continuous observables can be estimated in terms of modulus of continuity and the decay rate of tail of Young tower. In the rest of the second chapter we study the relations between the rates of decay of correlations for smooth observables and continuous observables. We show that if the rates of decay of correlations is known for observables () then it is possible to obtain decay of correlations for continuous observables in terms of modulus of continuity