arXiv subjects
Publications and source records attributed to Douglas Coates.
We consider one dimensional maps with several neutral fixed points that do not admit any physical measures. We show that there is simplex of measures so that every measure in this simplex has a basin which has full Hausdorff dimension.
We prove global-local mixing for a large class of dynamical systems with infinite invariant measure. In particular, we treat intermittent maps including maps with multiple neutral fixed points, nonMarkovian intermittent maps, and multidimensional nonMarkovian intermittent maps. We also prove global-local mixing for parabolic rational maps of the complex plane.
In this article we show that a large class of infinite measure preserving dynamical systems that do not admit physical measures nevertheless exhibit strong statistical properties. In particular, we give sufficient conditions for existence of a distinguished natural measure $\nu$ such that the pushforwards of any absolutely continuous probability measure converge to $\nu$. Moreover, we obtain a distributional limit law for empirical measures. We also extend existing results on the characterisation of the set of almost sure limit points for empirical measures. Our results apply to various intermittent maps with multiple neutral fixed points preserving an infinite $\sigma$-finite absolutely continuous measure.
We study a class $\widehat{\mathfrak{F}}$ of one-dimensional full branch maps introduced in [Doubly Intermittent Full Branch Maps with Critical Points and Singularities; D. Coates, S. Luzzatto, M. Mubarak, 2022], admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that $\widehat{\mathfrak{F}}$ can be partitioned into 3 pairwise disjoint subfamilies $$\widehat{\mathfrak{F}} = \mathfrak{F} \cup \mathfrak{F}_\pm \cup \mathfrak{F}_*$$ such that all $g \in \mathfrak{F}$ have a unique physical measure equivalent to Lebesgue, all $g \in \mathfrak{F}_{\pm}$ have a physical measure which is a Dirac-$\delta$ measure on one of the (repelling) fixed points, and all $g \in \mathfrak{F}_{*}$ are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are intermingled: they can all be approximated by maps in the other subfamilies in natural topologies.
We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild assumptions we prove the existence of a unique invariant mixing absolutely continuous probability measures, study its rate of decay of correlation and prove a number of limit theorems.
We consider perturbations of interval maps with indifferent fixed points, which we refer to as wobbly interval intermittent maps, for which stable laws for general H\"older observables fail. We obtain limit laws for such maps and H\"older observables. These limit laws are similar to the classical semistable laws previously established for random processes, but certain limitations imposed by the current dynamical set up are reflected in the main result. One of the considered examples is an interval map with a countable number of discontinuities, and to analyse it we need to construct a Markov/Young tower.