Dimension groups for interval maps II: the transitive case

Abstract

Abstract. Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any transitive piecewise monotonic map, and determines any transitive unimodal map up to conjugacy. For a transitive piecewise monotonic map which is not essentially injective, the associated dimension group is a direct sum of simple dimension groups, each with a unique state.

Dimension groups associated to $\beta$-expansions

“…The sofic case was solved in [21,Proposition 10.6], and the nonsofic case is got by comparing the previous theorem to [21, Proposition 10.5].…”

“…The sofic case was solved in [7,Corollary 15.3], and the non-sofic case is got by comparing the dimension group in the previous theorem, along with the distinguished order unit, to that of [21,Proposition 10.6].…”

“…The second construction, introduced in [20] and studied further in [21], is a special case of a direct construction valid for any interval map, specialized to the map T β defined above.…”

“…By [21,Proposition 10.6] the two constructions give the same dimension groups when the β-shift is sofic (cf. [13]).…”

“…[13]). This is seen by comparing explicit computations of the dimension groups in [21,Proposition 10.3] and [12,Theorem 6.1]. But since the dimension group of the fixed point algebras of the C * -algebras associated by Matsumoto to the β-shift X β is only determined as a group in [12], the problem of determining whether the dimension groups coincide in general has been left open.…”