arXiv subjects

I. Melbourne

Publications and source records attributed to I. Melbourne.

6 recordsLinked to original sources

arXiv2018

Explicit coupling argument for nonuniformly hyperbolic transformations

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. Here it is verified that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For nonuniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated to the inducing time. Finally, for nonuniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

math.DS↗

arXiv2017

Martingale-coboundary decomposition for families of dynamical systems

We prove statistical limit laws for sequences of Birkhoff sums of the type $\sum_{j=0}^{n-1}v_n\circ T_n^j$ where $T_n$ is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family $T_n$ is replaced by a fixed transformation $T$, and which is particularly effective in the case when $T_n$ varies with $n$. In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family $T_n$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards. As an application, we prove a homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

math.DS↗

arXiv2017

Existence and smoothness of the stable foliation for sectional hyperbolic attractors

We prove the existence of a contracting invariant topological foliation in a full neighborhood for partially hyperbolic attractors. Under certain bunching conditions it can then be shown that this stable foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the stable foliation for the classical Lorenz equation (and nearby vector fields) is better than $C^1$ which is crucial for recent results on exponential decay of correlations. In fact the foliation is at least $C^{1.278}$.

math.DS↗

arXiv2016

Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

We consider families of fast-slow skew product maps of the form \begin{align*} x_{n+1} = x_n+εa(x_n,y_n,ε), \quad y_{n+1} = T_εy_n, \end{align*} where $T_ε$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as $ε\to0$. Similar results are obtained also for continuous time systems \begin{align*} \dot x = εa(x,y,ε), \quad \dot y = g_ε(y). \end{align*} Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.

math.DS↗

arXiv2015

Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps

We prove that every geometric Lorenz attractor has superpolynomial decay of correlations with respect to the unique SRB measure. Moreover, we prove the Central Limit Theorem and Almost Sure Invariance Principle for the time-1 map of the flow of geometric Lorenz attractors. In particular, our results apply to the classical Lorenz attractor.

math.DS↗

arXiv2015

A Note on Diffusion Limits of Chaotic Skew Product Flows

We provide an explicit rigorous derivation of a diffusion limit - a stochastic differential equation with additive noise - from a deterministic skew-product flow. This flow is assumed to exhibit time-scale separation and has the form of a slowly evolving system driven by a fast chaotic flow. Under mild assumptions on the fast flow, we prove convergence to a stochastic differential equation as the time-scale separation grows. In contrast to existing work, we do not require the flow to have good mixing properties. As a consequence, our results incorporate a large class of fast flows, including the classical Lorenz equations.

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