Class-closing factor codes and constant-class-to-one factor codes from shifts of finite type

Abstract

Abstract. We define class-closing factor codes from shifts of finite type and show that they are continuing if their images are of finite type. We establish several relations between class-closing factor codes, continuing factor codes and constant-class-to-one factor codes. In particular it is shown that a factor code between irreducible shifts of finite type is constant-class-to-one if and only if it is bi-class-closing, generalising a result of Nasu.

Fiber-Mixing Codes between Shifts of Finite Type and Factors of Gibbs Measures

“…As constant-class-to-one codes are bi-continuing [7], Theorems 1 and 2 also strengthen the main results of [11], in which the existence of an infinite-to-one factor code implies the existence of a bi-continuing factor code.…”

“…For further details on symbolic dynamics, see [5]. Properties of class degree and transition classes can be found in [6][7][8].…”

“…We recall the properties of transition classes and class degrees. Details can be found in [6][7][8].…”

“…The following properties of factor codes were defined in [7,9], respectively. The definition of a fiber-mixing code is rather simple and can be stated without transition classes (see Figure 1).…”

“…This number is called the class degree of π. Properties of class degree and the structure of fibers and transition classes show that class degree may be regarded as a natural generalization of the degree to not necessarily finite-to-one factor codes [6][7][8].…”