Natural Extensions and Mixing for Semi-Group Actions
Metric properties of generalized Cantor products
On strong uniform distribution, II. The infinite-dimensional case
We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or , using the entropy method. It follows that such a chain with positive lower density is bad for . There also exist such bad chains with zero density.
A non-regular Toeplitz flow with preset pure point spectrum
Given an arbitrary countable subgroup of the torus, containing infinitely many rationals, we construct a strictly ergodic 0-1 Toeplitz flow with pure point spectrum equal to . For a large class of Toeplitz flows certain eigenvalues are induced by eigenvalues of the flow Y which can be seen along the aperiodic parts.
Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows
Let be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of is Borel isomorphic to an almost 1-1 extension of . Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz...
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.
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...
Spectral isomorphisms of Morse flows
A combinatorial description of spectral isomorphisms between Morse flows is provided. We introduce the notion of a regular spectral isomorphism and we study some invariants of such isomorphisms. In the case of Morse cocycles taking values in , where p is a prime, each spectral isomorphism is regular. The same holds true for arbitrary finite abelian groups under an additional combinatorial condition of asymmetry in the defining Morse sequence, and for Morse flows of rank one. Rank one is shown to...
A criterion for Toeplitz flows to be topologically isomorphic and applications
A dynamical system is said to be coalescent if its only endomorphisms are automorphisms. The question whether there exist coalescent ergodic dynamical systems with positive entropy has not been solved so far and it seems to be difficult. The analogous problem in topological dynamics has been solved by Walters ([W]). His example, however, is not minimal. In [B-K2], a class of strictly ergodic (hence minimal) Toeplitz flows is presented, which have positive entropy and trivial topological centralizers...
Local perturbations of energy and Kac's return time theorem.
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