Chains, entropy, coding

Abstract

Abstract. Various definitions of the entropy for countable-state topological Markov chains are considered. Concrete examples show that these quantities do not coincide in general and can behave badly under nice maps. Certain restricted random walks which arise in a problem in magnetic recording provide interesting examples of chains. Factors of some of these chains have entropy equal to the growth rate of the number of periodic orbits, even though they contain no subshifts of finite type with positive entropy; others are almost sofic-they contain subshifts of finite type with entropy arbitrarily close to their own. Attempting to find the entropies of such subshifts of finite type motivates the method of entropy computation by loop analysis, in which it is not necessary to write down any matrices or evaluate any determinants. A method for variable-length encoding into these systems is proposed, and some of the smaller subshifts of finite type inside these systems are displayed.

Classification problems for shifts on modules over a principal ideal domain

“…Recently, however, there has been some interest in considering also more general infinite alphabets. This type of generalization is a natural one to make and it comes up in various fields such as differentiable dynamics [4], nonuniqueness of equilibrium states in statistical mechanics [11], formal languages and automata [1,2], and also in more applicative contexts, such as coding for magnetic resonances [17]. In the mentioned papers Markovian shifts over countable alphabets are considered and questions like recurrence, topological entropy, ergodic measures are addressed.…”

On the Capacity of the Precision-Resolution System

“…It should be noted that Theorem 4 (as well as its large-alphabet counterpart, see Section IV-A) is a special case of [14,Theorem 7.5]. However, we choose to prove a different generalization which is given in Theorem 12.…”

Computing the topological entropy of shifts

“…However, it can be shown (see [14,Example 7.2.8], [18]) that there exist countable, irreducible covers of the full shift {0, 1} Z with Perron value λ < 2, hence log λ < h ({0, 1} Z ). Thus the logarithm of the Perron value need not be equal to the topological entropy of the corresponding coded system X.…”

“…But how to compute the uniform synchronization length? To solve this problem, the following definition is crucial, see [18].…”