ω\omega-recurrence in cocycles

After relating the notion of $\omega$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First, the generic situation is studied and shown to be $1/n$-recurrent. It is then shown that for any $\omega(n) <n^{-\epsilon}$, where $\epsilon>1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $\omega$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over interval exchange transformations of periodic type

Structure and evolution of strange attractors in non-elastic triangular billiards

We study pinball billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls are non-elastic: the outgoing angle with the normal vector to the boundary is a uniform factor $\lambda < 1$ smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter $\lambda$ is varied. For $\lambda$ in the interval (0, 1/3), we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of $\lambda$ the billiard dynamics gives rise to nonaccessible regions in phase space. For $\lambda$ close to 1, the attractor splits into three transitive components, the basins of attraction of which have fractal basin boundaries.Comment: 12 pages, 10 figures; submitted for publication. One video file available at http://sistemas.fciencias.unam.mx/~dsanders

Random walks on the torus with several generators

Our paper gives bounds for the rate of convergence for a class of random walks on the d-dimensional torus generated by a set of n vectors in R^d/Z^d. We give bounds on the discrepancy distance from Haar measure; our lower bound holds for all such walks, and if the generators arise from the rows of a "badly approximable" matrix, then there is a corresponding upper bound. The bounds are sharp for walks on the circle.Comment: 10 pages; related work at http://www.math.hmc.edu/~su/papers.htm

A theory of non-local linear drift wave transport

Transport events in turbulent tokamak plasmas often exhibit non-local or non-diffusive action at a distance features that so far have eluded a conclusive theoretical description. In this paper a theory of non-local transport is investigated through a Fokker-Planck equation with fractional velocity derivatives. A dispersion relation for density gradient driven linear drift modes is derived including the effects of the fractional velocity derivative in the Fokker-Planck equation. It is found that a small deviation (a few percent) from the Maxwellian distribution function alters the dispersion relation such that the growth rates are substantially increased and thereby may cause enhanced levels of transport.Comment: 22 pages, 2 figures. Manuscript submitted to Physics of Plasma

Optimal Polynomial Recurrence

Let $P\in\Z[n]$ with $P(0)=0$ and \VE>0. We show, using Fourier analytic techniques, that if N\geq \exp\exp(C\VE^{-1}\log\VE^{-1}) and $A\subseteq\{1,\...,N\}$, then there must exist $n\in\N$ such that \frac{|A\cap (A+P(n))|}{N}>(\frac{|A|}{N})^2-\VE. In addition to this we also show, using the same Fourier analytic methods, that if $A\subseteq\N$, then the set of \emph{\VE-optimal return times} R(A,P,\VE)=\{n\in \N \,:\,\D(A\cap(A+P(n)))>\D(A)^2-\VE\} is syndetic for every \VE>0. Moreover, we show that R(A,P,\VE) is \emph{dense} in every sufficiently long interval, in the sense that there exists an L=L(\VE,P,A) such that |R(A,P,\VE)\cap I| \geq c(\VE,P)|I| for all intervals $I$ of natural numbers with $|I|\geq L$ and c(\VE,P)=\exp\exp(-C\,\VE^{-1}\log\VE^{-1}).Comment: Short remark added and typos fixe

A Fractional Fokker-Planck Model for Anomalous Diffusion

In this paper we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality observing the transition from a Gaussian distribution to a L\'evy distribution. The statistical properties of the distribution functions are assessed by a generalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior.Comment: 22 pages, 14 figure

Rational approximation and arithmetic progressions

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed

First passage time exponent for higher-order random walks:Using Levy flights

We present a heuristic derivation of the first passage time exponent for the integral of a random walk [Y. G. Sinai, Theor. Math. Phys. {\bf 90}, 219 (1992)]. Building on this derivation, we construct an estimation scheme to understand the first passage time exponent for the integral of the integral of a random walk, which is numerically observed to be $0.220\pm0.001$. We discuss the implications of this estimation scheme for the $n{\rm th}$ integral of a random walk. For completeness, we also address the $n=\infty$ case. Finally, we explore an application of these processes to an extended, elastic object being pulled through a random potential by a uniform applied force. In so doing, we demonstrate a time reparameterization freedom in the Langevin equation that maps nonlinear stochastic processes into linear ones.Comment: 4 figures, submitted to PR

The Duffin-Schaeffer Conjecture with extra divergence II

This paper takes a new step in the direction of proving the Duffin-Schaeffer Conjecture for measures arbitrarily close to Lebesgue. The main result is that under a mild `extra divergence' hypothesis, the conjecture is true.Comment: 7 page

On the frequency of partial quotients of regular continued fractions

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are given by a modified variational principle.Comment: Accepted by Mathematical Proceedings of the Cambridge Philosophical Societ