Multiplicity of measures under factor codes and class degree joinings

2015/01/08 by Jisang Yoo, Yoo, Jisang · 1 citation
Mathematics · #37B10 (Primary) #37D35 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37B10 #msc:37D35

paper · pdf · doi:10.48550/arxiv.1501.01751

arxiv created 2015/01/08 · arxiv updated 2015/01/09

Abstract

Given a finite-to-one factor code π: X → Y between irreducible sofic shifts and an ergodic ν on Y with full support, it is known that the fiber π^-1_*(ν) has at most d_π ergodic measures in it where d_π is the degree of π. We introduce the notion of multiplicity for ergodic measures on X (that depends on π) and we prove that d_π is the sum of the multiplicity of μ where μ runs over the ergodic measures in π^-1_*(ν). We also build an appropriate generalization to infinite-to-one factor codes in relation to class degree and relatively maximal measures. We also define the notion of degree joining (for finite-to-one factor codes) and class degree joining (for infinite-to-one factor codes) which are the main tool for establishing our results

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