A. Iwanik

Absolutely continuous cocycles over irrational rotations

AbstractFor homeomorphisms

Code length between Markov processes

\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)

Integral Representations of Stochastic Kernels

(z, w ∈S1,α is irrational,ϕ:S1→S1) of the torusS1×S1 it is proved thatTϕ has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions wheneverϕ is absolutely continuous with nonzero topological degree and the derivative ofϕ is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.

A large set containing few orbits of measure preserving transformations

Abstract The existence of roots of stochastic operators on L 1 -spaces is considered. A necessary condition for the existence of n th roots of an ergodic and conservative operator is formulated in terms of eigenvalues (Theorem 1), extending known results for measure preserving transformations. Various conditions for roots of doubly stochastic operators are considered in Section 3. Two special classes of doubly stochastic operators are more closely analyzed: convolution operators (Sect. 5) and finite matrices (Sect. 6). A structure theorem for semigroups of singular doubly stochastic matrices is obtained (Theorem 6).

EMBEDDING SEMIGROUPS IN SEMIGROUPS GENERATED BY IDEMPOTENTS

LetX1 andX2 be two mixing Markov shifts over finite alphabet. If the entropy ofX1 is strictly larger than the entropy ofX2, then there exists a finitary homomorphism ϕ:X1→X2 such that the code length is anLp random variable for allp<4/3. In particular, the expected length of the code ϕ is finite.

INDEPENDENCE AND SCRAMBLED SETS FOR CHAOTIC MAPPINGS

We prove that an irrational number a admits a rational approximation la p/ql = o(f(q)) iff the irrational rotation Tx = {x + a} admits cyclic approximation with speed o(f(n)) . As an application to earlier results we obtain that a generic Anzai skew product over every irrational rotation is rank-i and for a.e. a most skew products admit cyclic approximation with speed o(1/n2 log n) .

Quasi-uniform convergence in compact dynamical systems

Publisher Summary This chapter focuses on the integral representations of stochastic kernels. It describes a Choquet-type representation. A similar representation by deterministic kernels in general fails in the convex set of Feller kernels on topological spaces, although it may be true for some nontrivial classes of kernels. A parallel representation problem in the measure preserving case has been considered by Sudakov. The chapter describes the analysis of ergodic properties. It is assumed that the stochastic kernels become transition probabilities, which can be composed and iterated. Integral representations are considered without necessarily assuming that the P ω are deterministic. Certain ergodic properties of the components P ω are preserved by the mixture P = ∫ P ω dλ(ω). This extends some earlier observations by Sine and Foguel on finite convex combinations of Markov and stochastic operators.

Norm attaining operators on Lebesgue spaces.

There exists a Borel set C of product Lebesgue measure one in the Hilbert cube having the property that, for every measure preserving transformationT of the unit interval, allT-orbits contained inC originate from a zero set. This settles an infinite dimensional version of a problem raised by Th. M. Rassias.