Given an irreducible sofic shift $X$ , we show that an irreducible shift of finite type $Y$ of lower entropy is a factor of $X$ if and only if it is a factor of $X$ by an open bi-continuing code. If these equivalent conditions hold and $Y$ is mixing, then any code from a proper subshift of $X$ to $Y$ can be extended to an open bi-continuing code on $X$ . These results are still valid when $X$ is assumed to be only an almost specified shift, i.e., a subshift satisfying an irreducible version of the specification property.

Given a factor code . In this paper, we investigate the structure of transition classes and present several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple, there cannot be a transition between two distinct transition classes over a right transitive point, answering a question raised by Quas.

The notion of topological entropy dimension for a $\mathbb{Z}$ -action has been introduced to measure the subexponential complexity of zero entropy systems. Given a ${\mathbb{Z}}^2$ -action, along with a ${\mathbb{Z}}^2$ -entropy dimension, we also consider a finer notion of directional entropy dimension arising from its subactions. The entropy dimension of a ${\mathbb{Z}}^2$ -action and the directional entropy dimensions of its subactions satisfy certain inequalities. We present several constructions of strictly ergodic ${\mathbb{Z}}^2$ -subshifts of positive entropy dimension with diverse properties of their subgroup actions. In particular, we show that there is a ${\mathbb{Z}}^2$ -subshift of full dimension in which every direction has entropy 0.

Given a code from a shift space to an irreducible sofic shift, any two of the three conditions—open, constant-to-one and (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.

Artículo de publicación ISIWe define class-closing factor codes from shifts of finite type and show that they are continuing if their domains are irreducible and 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, generalizing a result of Nasu.FONDECYT 11140369 313071

Given a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.