We study the simplex $\mathcal{M}_1(B)$ of probability measures on a Bratteli diagram $B$ which a... more We study the simplex $\mathcal{M}_1(B)$ of probability measures on a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation. We prove a criterion of unique ergodicity of a Bratteli diagram. In case when a finite rank $k$ Bratteli diagram $B$ has $l \leq k$ ergodic invariant measures, we describe the structures of the diagram and the subdiagrams which support these measures. We find conditions under which the extension of a measure from a uniquely ergodic subdiagram is a finite ergodic measure.
In this chapter, we collect definitions and some basic facts about the underlying spaces, endomor... more In this chapter, we collect definitions and some basic facts about the underlying spaces, endomorphisms, measurable partitions, etc., which are used throughout the book. Though these notions are known in ergodic theory, we discuss them for the reader’s convenience.
We present a unified study a class of positive operators called (generalized) transfer operators,... more We present a unified study a class of positive operators called (generalized) transfer operators, and of their applications to the study of endomorphisms, measurable partitions, and Markov processes, as they arise in diverse settings. We begin with the setting of dynamics in standard Borel, and measure, spaces.
In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isomet... more In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isometry operator of a Hilbert space in a unitary part and a unilateral shift.
We study the simplex $\mathcal{M}_1(B)$ of probability measures on a Bratteli diagram $B$ which a... more We study the simplex $\mathcal{M}_1(B)$ of probability measures on a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation. We prove a criterion of unique ergodicity of a Bratteli diagram. In case when a finite rank $k$ Bratteli diagram $B$ has $l \leq k$ ergodic invariant measures, we describe the structures of the diagram and the subdiagrams which support these measures. We find conditions under which the extension of a measure from a uniquely ergodic subdiagram is a finite ergodic measure.
In this chapter, we collect definitions and some basic facts about the underlying spaces, endomor... more In this chapter, we collect definitions and some basic facts about the underlying spaces, endomorphisms, measurable partitions, etc., which are used throughout the book. Though these notions are known in ergodic theory, we discuss them for the reader’s convenience.
We present a unified study a class of positive operators called (generalized) transfer operators,... more We present a unified study a class of positive operators called (generalized) transfer operators, and of their applications to the study of endomorphisms, measurable partitions, and Markov processes, as they arise in diverse settings. We begin with the setting of dynamics in standard Borel, and measure, spaces.
In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isomet... more In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isometry operator of a Hilbert space in a unitary part and a unilateral shift.
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Papers by Sergey Bezuglyi