Cantor aperiodic systems and Bratteli diagrams

Medynets, Konstantin

doi:10.1016/j.crma.2005.10.024

Cited by 55 publications

(56 citation statements)

References 8 publications

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“…Proof: It is well known [14][15][16] that the continuity of t and the minimality of φ determine a partition of K in clopen, disjoint subsets. Indeed, for every integer n ≥ 1 we set…”

Section: Kakutani-rohlin Partitionsmentioning

confidence: 98%

Linking invariants for smooth minimal solenoids

Arriagada

2015

Dynamical Systems

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In this paper we consider smooth diffeomorphisms of the 2-disk which are the identity on the boundary. We assume that the dynamics of any such a diffeomorphism φ restricted to a φ-invariant Cantor set is minimal and uniquely ergodic. Then the average linking number of the orbits of φ can be computed in two standard ways. We prove that the asymptotic average of the diagonal component of the Calabi invariant coincides with the Ruelle invariant of the minimal Cantor system.

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“…Proof: It is well known [14][15][16] that the continuity of t and the minimality of φ determine a partition of K in clopen, disjoint subsets. Indeed, for every integer n ≥ 1 we set…”

Section: Kakutani-rohlin Partitionsmentioning

confidence: 98%

Linking invariants for smooth minimal solenoids

Arriagada

2015

Dynamical Systems

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“…It is clear that if T is minimal then every point of X is a basic set. It was proved in [Med06] that every aperiodic Cantor system (X, T ) has a basic set. This is a crucial step in the proof of the following theorem.…”

Section: Example Of a Bratteli Diagrammentioning

confidence: 99%

“…Theorem 2.11. [Med06] Let (X, T ) be a Cantor aperiodic system with a basic set Y . There exists an ordered Bratteli diagram B = (V, E, ω) such that (X, T ) is conjugate to a Bratteli-Vershik dynamical system (X B , ϕ ω ).…”

Section: Example Of a Bratteli Diagrammentioning

confidence: 99%

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Invariant measures for Cantor dynamical systems

Bezuglyi¹,

Karpel²

2020

Dynamics: Topology and Numbers

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This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system (X, T ) can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.1991 Mathematics Subject Classification. Primary 37A05, 37B05; Secondary 28D05, 28C15.

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“…Subsequently, there have been numerous studies on the Bratteli-Vershik models. For instance, Medynets [Med06] showed that aperiodic Cantor systems admit Bratteli-Vershik models, where the basic sets (see Definition 3.17) coincide with the sets of maximal (minimal) paths of the ordered Bratteli diagrams. This discovery diversified the study of Bratteli-Vershik models whose basic sets are not single points.…”

Section: Introductionmentioning

confidence: 99%

Bratteli–Vershik models and graph covering models

Shimomura

2020

Advances in Mathematics

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Based on our previous graph covering method, we introduce weighted graph covering models and flexible graph covering models that are almost equivalent to the well-known Bratteli-Vershik models. These models play important roles in showing that every invertible dynamical system on compact metrizable zero-dimensional space admits a non-trivial Bratteli-Vershik model and a basic set. We can also obtain an analogue of Krieger's lemma for compact metrizable zero-dimensional systems. The flexible graph covering models enable us to consider "stationary" graph covering models, by which some portion of the substitution subshifts can be expressed. As an application, we show a way of constructing some class of transitive substitution subshifts.

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Cantor aperiodic systems and Bratteli diagrams

Cited by 55 publications

References 8 publications

Linking invariants for smooth minimal solenoids

Linking invariants for smooth minimal solenoids

Invariant measures for Cantor dynamical systems

Bratteli–Vershik models and graph covering models

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