Smooth Non-Bernoulli K-Automorphisms.
Sofic Systems and Graphs.
Solenoidal Automorphisms with Specifications.
Solvability of the functional equation f = (T-I)h for vector-valued functions
Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and in measure for some f ∈ M(μ;X), then also in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X) satisfying...
Some combinatorial problems on the measurability of functions with respect to invariant extensions of the Lebesgue measure
Some constructions of strictly ergodic non-regular Toeplitz flows
We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.
Some limit ratio theorem related to a real endomorphism in case of a neutral fixed point
Some new cases of realization of spectral multiplicity function for ergodic transformations
Some New Examples of Gibbs Measures.
Some properties od step-punctions connected with extensions of measures
Some Remarks Concerning an Example of a Minimal, Non-Uniquely Ergodic Interval Exchange Transformation.
Some remarks on the metrical transitivity of invariant and quasi-invariant measures
Some Results on Affine Transformations of Compact Groups.
Some theorems concerning the class of weight functions in the transformation theory for mesure space
Some thoughts about Segal's ergodic theorem
Over fifty years ago, Irving Segal proved a theorem which leads to a characterization of those orthogonal transformations on a Hilbert space which induce ergodic transformations. Because Segal did not present his result in a way which made it readily accessible to specialists in ergodic theory, it was difficult for them to appreciate what he had done. The purpose of this note is to state and prove Segal's result in a way which, I hope, will win it the recognition which it deserves.
Spectral properties of ergodic dynamical systems conjugate to their composition squares
Let S and T be automorphisms of a standard Borel probability space. Some ergodic and spectral consequences of the equation ST = T²S are given for T ergodic and also when Tⁿ = I for some n>2. These ideas are used to construct examples of ergodic automorphisms S with oscillating maximal spectral multiplicity function. Other examples illustrating the theory are given, including Gaussian automorphisms having simple spectra and conjugate to their squares.
Spectral properties of G-symbolic Morse shifts
Spectral properties of skew-product diffeomorphisms of tori
Spectrum of multidimensional dynamical systems with positive entropy
Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov -action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to -actions. Next, using its relative version, we extend to -actions some other general results connecting spectrum and entropy.
Spontaneous clustering in theoretical and some empirical stationary processes*
In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper...