Structure of transition classes for factor codes on shifts of finite type

Abstract

Artículo de publicación ISIGiven a factor code pi from a shift of finite type X onto a sofic shift Y, the class degree of pi is defined to be the minimal number of transition classes over the points of Y. 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.Fondecyt; Basic Science Research Program through the National Research Foundation of Korea (NRF) - Ministry of Educatio

Relative equilibrium states and class degree

“…Since w is a transition block, u is routable through at least one member of M at time n. To show that u is routable through at most one member of M at time n, we suppose to the contrary that u is routable through two distinct members a (1) and a (2) By Poincare's recurrence theorem, for µ-almost every point x in the cylinder [u] ⊂ X, the block u occurs infinitely many times to the right in x. And for µ-almost every point x ∈ X, the point π(x) has exactly d transition classes over it.…”

“…, µ d+1 are distinct ergodic measures of relative maximal entropy over ν where d is the number of letters (for X) that project to a fixed letter b for Y with ν(b) > 0. Form a relatively independent joining of the d + 1 measures over ν. Pigeonhole's principle then forces at least two, say µ 1 , µ 2 , of the d + 1 measures to have the property λ({(x (1) , x (2) ) :…”

“…i , one can construct a point x (3) which is the result of splicing parts of x (1) or x (2) depending on the outcome of tossing a fair coin at every i for which x (1) i = x (2) i . The probability distribution of the new point x (3) is a measure µ 3 on X which projects to ν.…”

“…The main difficulty in applying Antonioli's observation to our setting is that we cannot restore the old points x (1) , x (2) from the new points x (3) , x (4) . But since µ 1 , µ 2 are distinct ergodic measures, long blocks from x (1) are distinguishable from long blocks of x (2) with low probability of error. The rate of error goes to zero as the blocks become longer.…”

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

“…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].…”