arXiv subjects

Christopher Bose

Publications and source records attributed to Christopher Bose.

11 recordsLinked to original sources

arXiv2020

Random Composition of L-S-V Maps Sampled Over Large Parameter Ranges

Liverani-Saussol-Vaienti (L-S-V) maps form a family of piecewise differentiable dynamical systems on $[0,1]$ depending on one parameter $\omega\in\mathbb R^+$. These maps are everywhere expanding apart from a neutral fixed point. It is well known that depending on the amount of expansion close to the neutral point, they have either an absolutely continuous invariant probability measure and polynomial decay of correlations ($\omega <1$), or a unique physical measure that is singular and concentrated at the neutral point ($\omega >1$). In this paper, we study the composition of L-S-V maps whose parameters are randomly sampled from a range in $\mathbb R^+$, and where these two contrasting behaviours are mixed. We show that if the parameters $\omega<1$ are sampled with positive probability, then the stationary measure of the random system is absolutely continuous; the annealed decay rate of correlations is close (or in some cases equal) to the fastest rate of decay among those of the sampled systems; and suitably rescaled Birkhoff averages converge to limit laws. In contrast to previous studies where $\omega \in [0,1]$, we allow $ \omega >1$ in our sampling distribution. We also show that one can obtain similar decay of correlation rates for $\omega \in [0,\infty)$, when sampling is done with respect to a family of smooth, heavy-tailed distributions.

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arXiv2017

Quenched decay of correlations for slowly mixing systems

We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of Liverani-Saussol-Vaienti maps with parameters in $[\alpha_0,\alpha_1]\subset (0,1)$ chosen independently with respect to a distribution $\nu$ on $[\alpha_0,\alpha_1]$ and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every $\delta >0$, for almost every $\omega \in [\alpha_0,\alpha_1]^\mathbb Z$, the upper bound $n^{1-\frac{1}{\alpha_0}+\delta}$ holds on the rate of decay of correlation for H\"older observables on the fibre over $\omega$. For three different distributions $\nu$ on $[\alpha_0,\alpha_1]$ (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from $n^{-\frac{1}{\alpha_0}}$ to $(\log n)^{\frac{1}{\alpha_0}}\cdot n^{-\frac{1}{\alpha_0}}$ to $(\log n)^{\frac{2}{\alpha_0}}\cdot n^{-\frac{1}{\alpha_0}}$ respectively.

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arXiv2016

On irreducibility of Oseledets subspaces

For a cocycle of invertible real $n$-by-$n$ matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of $\mathbb{R}^n$; that is, above each point in the base space, $\mathbb{R}^n$ is written as a direct sum of equivariant subspaces, one for each Lyapunov exponent of the cocycle. It is natural to ask if these summands may be further decomposed into equivariant subspaces; that is, if the Oseledets subspaces are reducible. We prove a theorem yielding sufficient conditions for irreducibility of the trivial equivariant subspaces $\mathbb{R}^2$ and $\mathbb{C}^2$ for $O_2(\mathbb{R})$-valued cocycles and give explicit examples where the conditions are satisfied.

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arXiv2015

Mixing rates and limit theorems for random intermittent maps

We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps $T_\alpha$ using the full parameter range $0< \alpha < \infty$, in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e.\ smallest $\alpha$) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of return time intervals for the \emph{quenched} dynamics. The main applications of this estimate are to \textit{limit laws} (in particular, CLT and stable laws, depending on the parameters chosen in the range $0 < \alpha < 1$) for the associated skew product; these are detailed in Theorem 3.2. Since our estimates in Theorem 1.1 also hold for $1 \leq \alpha < \infty$ we study a piecewise affine version of our random transformations, prove existence of an infinite ($\sigma-$finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.

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arXiv2013

Rigorous pointwise approximations for invariant densities of nonuniformly expanding maps

We use an Ulam-type discretization scheme to provide pointwise approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate $C^*\cdot\frac{\ln m}{m}$, where $C^*$ is a computable fixed constant and $m^{-1}$ is the mesh size of the discretization.

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arXiv2013

Decay of correlation for random intermittent maps

We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate.

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arXiv2013

Numerical approximation of conditionally invariant measures via Maximum Entropy

It is well known that open dynamical systems can admit an uncountable number of (absolutely continuous) conditionally invariant measures (ACCIMs) for each prescribed escape rate. We propose and illustrate a convex optimisation based selection scheme (essentially maximum entropy) for gaining numerical access to some of these measures. The work is similar to the Maximum Entropy (MAXENT) approach for calculating absolutely continuous invariant measures of nonsingular dynamical systems, but contains some interesting new twists, including: (i) the natural escape rate is not known in advance, which can destroy convex structure in the problem; (ii) exploitation of convex duality to solve each approximation step induces important (but dynamically relevant and not at first apparent) localisation of support; (iii) significant potential for application to the approximation of other dynamically interesting objects (for example, invariant manifolds).

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arXiv2012

Polynomial decay of correlations in the generalized baker's transformation

We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Holder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is non-uniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simple geometry in the construction. For this reason we propose this class of maps as an excellent testing ground for new techniques for the analysis of decay of correlations in non-uniformly hyperbolic systems. Finally, some of our examples can be seen to be extensions of certain 1-D non-uniformly expanding maps that have appeared in the literature over the last twenty years thereby providing a unified treatment of these interesting and well-studied examples.

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arXiv2012

Ulam's method for Lasota-Yorke maps with holes

Ulam's method is a rigorous numerical scheme for approximating invariant densities of dynamical systems. The phase space is partitioned into connected sets and an inter-set transition matrix is computed from the dynamics; an approximate invariant density is read off as the leading left eigenvector of this matrix. When a hole in phase space is introduced, one instead searches for \emph{conditional} invariant densities and their associated escape rates. For Lasota-Yorke maps with holes we prove that a simple adaptation of the standard Ulam scheme provides convergent sequences of escape rates (from the leading eigenvalue), conditional invariant densities (from the corresponding left eigenvector), and quasi-conformal measures (from the corresponding right eigenvector). We also immediately obtain a convergent sequence for the invariant measure supported on the survivor set. Our approach allows us to consider relatively large holes. We illustrate the approach with several families of examples, including a class of Lorenz maps.

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arXiv2010

Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty}$-norm

In this article we study a piecewise linear discretization schemes for transfer operators (Perron-Frobenius operators) associated with interval maps. We show how these can be used to provide rigorous {\bf pointwise} approximations for invariant densities of Markov interval maps. We also derive the order of convergence of the approximate invariant density to the real one in the $L^{\infty}$-norm. The outcome of this paper complements rigorous results on $L^1$ approximations of invariant densities \cite{KMY} and recent results on the formulae of escape rates of open dynamical systems \cite{KL2}. We implement our computations on two examples (one rigorous and one non-rigorous) to illustrate the feasibility and efficiency of our schemes.

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arXiv2009

Quasi-Invariant measures, escape rates and the effect of the hole

Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number $\ell$, $0<\ell<1$, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than $-\ln(1-\ell)$. The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.

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