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Topological Boundaries for Countable State Markov Shifts
Author(s) -
Fiebig Doris,
Fiebig UlfRainer
Publication year - 1995
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-70.3.625
Subject(s) - mathematics , markov chain , locally compact space , countable set , boundary (topology) , discrete mathematics , pure mathematics , compact space , transitive relation , metric space , second countable space , topology (electrical circuits) , combinatorics , mathematical analysis , statistics
Let ( S , σ) be a transitive locally compact countable state Markov shift. If ( Ŝ , σ ^ ) is some compact metric dynamical system (‘cmds’) that contains ( S , σ) as a dense subsystem then the space S * = Ŝ − S together with the homeomorphismσ * = σ ^ | S *will be called a boundary for ( S , σ). Any cmds ( X, T ) is shown to be a boundary for some transitive locally compact countable state Markov shift (Theorem A). We define a new canonical compactification whose boundary is a conjugacy invariant that distinguishes Markov shifts according to their behaviour ‘at infinity’ (Theorem B). This boundary, together with a periodic point condition, will be used to characterize the set of all chain‐recurrent boundaries of a given transitive locally compact countable state Markov shift (Theorems C, D).