Strong approximation of partial sums under dependence conditions with application to dynamical systems

In this paper, we obtain precise rates of convergence in the strong invariance principle for stationary sequences of real-valued random variables satisfying weak dependence conditions including strong mixing in the sense of Rosenblatt (1956) [30] as a special case. Applications to unbounded functions of intermittent maps are given.     

Some simple conditions implying topological transitivity for interval maps

Summary. We provide some sufficient conditions for topological transitivity of piecewise monotonic maps on [0,1]. Our theorems provide shorter and elementary proofs for some known recent results.     

From rates of mixing to recurrence times via large deviations

A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. In this paper we show that in many cases stochastic-like behaviour itself implies that the system has certain non-trivial geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration. As a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include Viana maps and multidimensional piecewise expanding maps.     

Geometry of expanding absolutely continuous invariant measures and the liftability problem

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs–Markov–Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.     

Measurable sensitivity

We introduce the notions of measurable and strong measurable sensitivity, which are measure-theoretic versions of the conditions of sensitive dependence on initial conditions and strong sensitive dependence on initial conditions, respectively. Strong measurable sensitivity is a consequence of light mixing, implies that a transformation has only finitely many eigenvalues, and does not exist in the infinite measure-preserving case. Unlike the traditional notions of sensitive dependence, measurable and strong measurable sensitivity carry up to measure-theoretic isomorphism, thus ignoring the behavior of the transformation on null sets and eliminating dependence on the choice of metric.

     

Distributional limit theorems in infinite ergodic theory

We present a unified approach to the Darling-Kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure. Our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator. We illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps. We also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations.     

连续自映射及其扭扩半流的不变测度

本文研究了紧致度量空间上连续自映射及连续半流的不变测度,并且证明了如下结论:(1)在拓扑等价的无不动点的连续半流的不变测度之间以及在连续自映射及其扭扩半流的不变测度之间存在一一对应;(2)作为(1)的应用,给出如下结论(见[2,定理2.1]):“环面上无不动点的连续流是唯一遍历的当且仅当它至多有一条周期轨”一个易接受的证明.     

Measurable dynamics of maps on profinite groups

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on _p.     

Chaotic sets of continuous and discontinuous maps

This paper discusses Li-Yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps. Scrambled sets and maximal scrambled sets are introduced to characterize Li-Yorke chaotic sets. The orbit invariant for a scrambled set is discussed. Some properties about maximality, equivalence and uniqueness of maximal scrambled sets are also discussed. It is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps, such as the Gauss map, interval exchange transformations, and a class of planar piecewise isometries, are studied. Finally, some open problems on scrambled sets are listed and remarked.     

On the Gray index of phantom maps

We study a homotopy invariant of phantom maps called the Gray index. In particular, it is conjectured that the Gray index of an essential phantom map between finite-type spaces is always finite. We obtain some partial results on this conjecture, using a tower-theoretic interpretation of the Gray index.     

Renormalization and conjugacy of piecewise linear Lorenz maps

We investigate the uniform piecewise linearizing question for a family of Lorenz maps. Let f be a piecewise linear Lorenz map with different slopes and positive topological entropy, we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy: it is either bi-Lipschitz or singular depending on whether f is renormalizable or not. f is renormalizable if and only if its rotation interval degenerates to be a rational point. Furthermore, if the endpoints are periodic points with the same rotation number, then the conjugacy is quasisymmetric.     

Renormalization for piecewise smooth homeomorphisms on the circle

In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy–Veech renormalizations of generalized interval exchange maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converge to the set of piecewise affine interval exchange maps.     

An analogue of the Harer-Zagier formula for unicellular maps on general surfaces

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is simply connected. In a famous article, Harer and Zagier established a formula for the generating function of unicellular maps counted according to the number of vertices and edges. The keystone of their approach is a counting formula for unicellular maps on orientable surfaces with n edges, and with vertices colored using every color in [q] (adjacent vertices are authorized to have the same color). We give an analogue of this formula for general (locally orientable) surfaces.Our approach is bijective and is inspired by Lass?s proof of the Harer-Zagier formula. We first revisit Lass?s proof and twist it into a bijection between unicellular maps on orientable surfaces with vertices colored using every color in [q], and maps with vertex set [q] on orientable surfaces with a marked spanning tree. The bijection immediately implies Harer-Zagier?s formula and a formula by Jackson concerning bipartite unicellular maps. It also shed a new light on constructions by Goulden and Nica, Schaeffer and Vassilieva, and Morales and Vassilieva. We then extend the bijection to general surfaces and obtain a correspondence between unicellular maps on general surfaces with vertices colored using every color in [q], and maps on orientable surfaces with vertex set [q]with a marked planar submap. This correspondence gives an analogue of the Harer-Zagier formula for general surfaces. We also show that this formula implies a recursion formula due to Ledoux for the numbers of unicellular maps with given numbers of vertices and edges.     

The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

Given a continuous dynamical system (X,T) with the specification property, and a sequence of asymptotically additive continuous functions, we consider the irregular set for it and show that this set is either empty or carries full asymptotically additive topological pressure.     

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ. We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.     

One-sided asymptotically mean stationary channels

This paper proposes an analysis of asymptotically mean stationary (AMS) communication channels. A hierarchy based on stability properties (stationarity, quasi-stationarity, recurrence and asymptotically mean stationarity) of channels is identified. Stationary channels are a subclass of quasi-stationary channels which are a subclass of recurrent AMS channels which are a subclass of AMS channels. These classes are proved to be stable under Markovian composition of channels (e.g., the cascade of AMS channels is an AMS channel). Characterizations of channels of each class are given. Some properties of the quasi-stationary mean of a channel are established. Finally, ergodicity conditions of AMS channels are gathered.     

On negative limit sets for one-dimensional dynamics

In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every α-limit set is an ω-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all α-limit sets of interval maps is not closed in the Hausdorff metric (and thus some ω-limit sets are never obtained as α-limit sets). Moreover, we prove that the set of all recurrent points is closed if and only if the space of all α-limit sets is closed.     

Möbius disjointness for topological models of ergodic measure-preserving systems with quasi-discrete spectrum

By using a decomposition result for ergodic measure-preserving system with quasi-discrete spectrum, we prove that a generic point of an ergodic quasi-discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that Sarnak's Möbius Disjointness Conjecture holds for any topological model of an ergodic measure-preserving system with quasi-discrete spectrum.     

Dynamical systems on lattices with decaying interaction II: Hyperbolic sets and their invariant manifolds

This is the second part of the work devoted to the study of maps with decay in lattices. Here we apply the general theory developed in Fontich et al. (2011) [3] to the study of hyperbolic sets. In particular, we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set, with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map. We also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds.     

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T  ; that there exist frequently hypercyclic operators on the sequence space _c0$c_0$ admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.