Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p>1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic–Lin and Wu–Woodroofe....

In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if $f\in L^p$, $p$ large enough, is a Wiener-Wintner function then, for all $\gamma \in (1+\frac{1}{2p}-\frac{\beta }{2},1]$, there exists a set $X_f$ of full measure for which the series ${\sum }_{n=1}^{\infty }\frac{f\left(T^{n}x\right)e^{2\pi inϵ}}{n^{\gamma }}$ converges uniformly with respect to $ϵ$.

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space ${}_{\phi }$ has pure discrete spectrum, we describe the collection ${}_{\phi }^P$ of pairs of proximal tilings in ${}_{\phi }$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then ${}_{\phi }$ and ${}_{\psi }$ are homeomorphic if and...

Let P be a Markov kernel on a measurable space E with countably generated σ-algebra, let w:E→[1, +∞[ such that Pw≤Cw with C≥0, and let ${ℬ}_w$ be the space of measurable functions onE satisfying ‖f‖w=sup{w(x)−1|f(x)|, x∈E}<+∞. We prove that Pis quasi-compact on $({ℬ}_w,\parallel ·{\parallel }_w)$ if and only if, for all $f\in {ℬ}_w$, ${\left(\frac{1}{n}{\sum }_{k=1}^n{P}^{k}f\right)}_n$ contains a subsequence converging in ${ℬ}_w$ toΠf=∑di=1μi(f)vi, where the vi’s are non-negative bounded measurable functions on E and the μi’s are probability distributions on E. In particular, when the space of...

It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f(kx-n)+{\sum }_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in \mathbb{Z}}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f(kx-y)dy$ has also various interesting applications....

We describe the natural framework in which the relative spectral theory is developed. We give some results and indicate how they relate to two open problems in ergodic theory. We also compute the relative entropy of gaussian extensions of ergodic transformations.

The Ruelle operator and the associated Perron-Frobenius property have been extensively studied in dynamical systems. Recently the theory has been adapted to iterated function systems (IFS) $(X,{w_j}_{j=1}^m,{p_j}_{j=1}^m)$, where the $w_j$’s are contractive self-maps on a compact subset $X\subseteq {ℝ}^d$ and the $p_j$’s are positive Dini functions on X [FL]. In this paper we consider Ruelle operators defined by weakly contractive IFS and nonexpansive IFS. It is known that in such cases, positive bounded eigenfunctions may not exist in general. Our theorems...

We study spectral properties of Anzai skew products $T_{\phi }:²\to ²$ defined by $T_{\phi }(z,\omega )=(e^{2\pi i\alpha }z,\phi \left(z\right)\omega )$, where α is irrational and φ: → is a measurable cocycle. Precisely, we deal with the case where φ is piecewise absolutely continuous such that the sum of all jumps of φ equals zero. It is shown that the simple continuous singular spectrum of $T_{\phi }$ on the orthocomplement of the space of functions depending only on the first variable is a “typical” property in the above-mentioned class of cocycles, if α admits a sufficiently fast approximation....

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.

The classical Banach principle is an essential tool for the investigation of ergodic properties of Cesàro subsequences. The aim of this work is to extend the Banach principle to the case of stochastic convergence in operator algebras. We start by establishing a sufficient condition for stochastic convergence (stochastic Banach principle). Then we prove stochastic convergence for bounded Besicovitch sequences, and as a consequence for uniform subsequences.

We show that the set of those Markov semigroups on the Schatten class ₁ such that in the strong operator topology $li{m}_{t\to \infty }T\left(t\right)=Q$, where Q is a one-dimensional projection, form a meager subset of all Markov semigroups.

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 $f$.

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.