arXiv subjects
Publications and source records attributed to Roland Zweimüller.
We establish abstract local limit theorems for hitting times and return-times of suitable sequences (A_{l}) of asymptotically rare events in ergodic probability preserving dynamical systems, including versions for tuples of consecutive times and positions of the hits. These results are shown to apply in the setup of Gibbs-Markov systems.
While routinely used in other areas of dynamics, image sets are ill-defined objects in general non-invertible measurable dynamics. We propose a way of consistently working with image sets of null-preserving (and hence, in particular, of measure-preserving) maps. This concept is illustrated in the context of basic ergodic properties like recurrence, ergodicity, exactness and existence of generators. It allows us to turn various suasive but logically false statements about set-theoretic images into actual theorems, and to eliminate extra assumptions on the measurability of images from some classical results.
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
We establish abstract limit theorems which provide sufficient conditions for a sequence $(A_{l})$ of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the $A_{l}$ to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to $A_{l}$ before some time $τ_{l}$ which is of order $o(1/μ(A_{l}))$. In particular, no assumptions on the asymptotic behavior of the system akin to classical mixing conditions are used. We also discuss some general questions about the asymptotic behaviour of spatial and spatiotemporal processes, and illustrate our results in a setup of simple prototypical systems.
For a class of locally (but not necessarily uniformly) Lipschitz continuous $d$-dimensional observables over a Gibbs-Markov system, we show that convergence of (suitably normalized and centered) ergodic sums to a non-Gaussian stable vector is equivalent to the distribution belonging to the classical domain of attraction, and that it implies a weak invariance principle in the (strong) Skorohod $\mathcal{J}_{1}$-topology on $\mathcal{D}([0,\infty),\mathbb{R}^{d})$. The argument uses the classical approach via finite-dimensional marginals and $\mathcal{J}_{1}$-tightness. As applications, we record a Spitzer-type arcsine law for certain $\mathbb{Z}% $-extensions of Gibbs-Markov systems, and prove an asymptotic independence property of excursion processes of intermittent interval maps.
We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $μ(E_{k})\rightarrow0$ in recurrent ergodic dynamical systems preserving an infinite measure $μ$. Some properties which are easy in finite measure situations break down in this null-recurrent setup. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-α$ with $α\in(0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge iff the corresponding hitting time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to $\mathcal{E}^{1/α}\,\mathcal{G}_α$, where $\mathcal{E}$ and $\mathcal{G}_α$ are independent random variables, with $\mathcal{E}$ exponentially distributed and $\mathcal{G}% _α$ following the one-sided stable law of order $α$ (and $\mathcal{G}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperblic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $α=0$ case.
We discuss limit distributions for hitting-time functions of certain exceptional families of asymptotically rare events for ergodic probability preserving transformations. The abstract core is an inducing argument. The latter applies, for example, to shrinking intervals around periodic points (both uniformly expanding and neutral) of certain finite measure preserving interval maps. In particular, we give a complete answer to a question raised in [FFTV].
We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of examples covered by our result are Pomeau-Manneville intermittency maps, where convergence for the induced system is in the standard Skorohod J_1 topology. For the full system, convergence in the J_1 topology fails, but we prove convergence in the M_1 topology.
We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as r goes to zero, of the return time to (-r,r). We then refine this result by establishing a limit theorem for the hitting-time distributions of (x-r,x+r) with arbitrary real x.
We prove distributional limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from non-integrable observables over certain piecewise expanding maps. This is done by extending Darling-Kac theory to a suitable family of infinite measure preserving transformations.
We study systems of globally coupled interval maps, where the identical individual maps have two expanding, fractional linear, onto branches, and where the coupling is introduced via a parameter - common to all individual maps - that depends in an analytic way on the mean field of the system. We show: 1) For the range of coupling parameters we consider, finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. 2) For the same range of parameters, the self-consistent Perron-Frobenius operator which captures essential aspects of the corresponding infinite-size system (arising as the limit of the above when the system size tends to infinity), undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium.