Open maps between shift spaces

Abstract

Abstract. Given a code from a shift space to an irreducible sofic shift, any two of the following three conditions -open, constant-to-one, (right or left) closing -imply the third. If the range is not sofic, then the same result holds when bi-closingness replaces closingness. Properties of open mappings between shift spaces are investigated in detail. In particular, we show that a closing open (or constant-to-one) extension preserves the structure of a sofic shift.

On the existence of open and bi-continuing codes

“…In this paper, we focus on two kinds of codes: Open codes and bi-continuing codes, i.e., left and right continuing codes. We will generalize the above results in both codes between shifts of finite type have been studied in several contexts [9,14,10]. A finite-to-one factor code between irreducible shifts of finite type is open if and only if it is bi-closing [14].…”

“…For more details on symbolic dynamics, see [12]. For a perspective for open maps between shift spaces, see [10].…”

“…[10] Let X be a shift of finite type and φ an open factor code from X to a shift space Y . Then Y is of finite type.…”

Topological conjugacy and its relations for symbolic matrices

“…(2) φ is bi-closing, (3) φ is an open map. Jung (2009Jung ( , 2011 also considers this problem and shows that for sofic shifts any two of the above three conditions implies the other. By introducing a numerical computation on entries of integral sub-matrices of sofic shifts, we determine whether each finite graph is right-or left-closing.…”

“…6. right-closing factor code in symbolic matrices Jung (2009) proved that for φ : X → Y is a factor code of two shifts of finite type with equal entropy; right-closing, open and constant-to-one factor codes are equivalent. In this section, we show the relation between the properties of factor code on symbolic matrices and factors on their integral sub-matrices.…”