We show that semisimple actions of l.c.s.c. Abelian groups and cocycles with values in such groups can be used to build new examples of semisimple automorphisms (ℤ-actions) which are relatively weakly mixing extensions of irrational rotations.

We study relationships between sequence entropy and the Kronecker and rigid algebras. Let (Y,,ν,T) be a factor of a measure-theoretical dynamical system (X,,μ,T) and S be a sequence of positive integers with positive upper density. We prove there exists a subsequence A ⊆ S such that $h_{\mu }^A(T,\xi |)=H_{\mu }\left(\xi \right|\left(X\right|Y))$ for all finite partitions ξ, where (X|Y) is the Kronecker algebra over . A similar result holds for rigid algebras over . As an application, we characterize compact, rigid and mixing extensions via relative sequence...

We prove density modulo $1$ of the sets of the form$$\{{\mu }^m{\lambda }^n\xi +r_m:n,m\in \mathbb{N}\},$$where $\lambda ,\mu \in ℝ$ is a pair of rationally independent algebraic integers of degree $d\ge 2,$ satisfying some additional assumptions, $\xi \ne 0,$ and $r_m$ is any sequence of real numbers.

We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...

We study the ergodicity of a multi-class queueing model via fluid limits which have the advantage of describing the model in macroscopic form. We consider a model of processing bandwidth requests. Our system is defined by a network of capacity C=N, and a queue which contains an infinite number of items of various sizes 1, a' and b' with 1 < a' < b' < N. The problem considered is: Under what conditions on the parameters of some large classes of networks, do they reach the stationary regime?...

In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably ergodic. Our main tools are the new concepts – julienne density point and julienne quasi-conformality of the stable and unstable holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points.

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition...

On a manifold X of dimension at least two, let μ be a nonatomic measure of full support with μ(∂X) = 0. The Oxtoby-Ulam Theorem says that ergodicity of μ is a residual property in the group of homeomorphisms which preserve μ. Daalderop and Fokkink have recently shown that density of periodic points is residual as well. We provide a proof of their result which replaces the dependence upon the Annulus Theorem by a direct construction which assures topologically robust periodic points.

We call a sequence $\left(T_n\right)$ of measure preserving transformations strongly mixing if $P(T_n^{-1}A\cap B)$ tends to $P\left(A\right)P\left(B\right)$ for arbitrary measurable $A$, $B$. We investigate whether one can pass to a suitable subsequence $\left(T_{n_k}\right)$ such that $\frac{1}{K}{\sum }_{k=1}^{K}f\left(T_{n_k}\right)⟶\int f\mathrm{d}P$ almost surely for all (or “many”) integrable $f$.

We consider S-unimodal Misiurewicz maps T with a flat critical point c and show that they exhibit ergodic properties analogous to those of interval maps with indifferent fixed (or periodic) points. Specifically, there is a conservative ergodic absolutely continuous σ-finite invariant measure μ, exact up to finite rotations, and in the infinite measure case the system is pointwise dual ergodic with many uniform and Darling-Kac sets. Determining the order of return distributions to suitable reference...

Let T be a positive linear contraction of $L_1$ of a σ-finite measure space (X,Σ,μ) which overlaps supports. In general, T need not be completely mixing, but it is in the following cases: (i) T is the Frobenius-Perron operator of a non-singular transformation ϕ (in which case complete mixing is equivalent to exactness of ϕ). (ii) T is a Harris recurrent operator. (iii) T is a convolution operator on a compact group. (iv) T is a convolution operator on a LCA group.

We prove the absence of mixing for special flows built over (1) an irrational rotation and under a function whose Fourier coefficients are of order O(1/|n|), and (2) an irrational rotation (satisfying a diophantine condition) and under a function having a finite number of singularities of a logarithmic type. These results generalize two theorems of Kochergin.