Abstract
We introduce the concept of inflation word entropy for random substitutions with a constant and primitive substitution matrix. Previous calculations of the topological entropy of such systems implicitly used this concept and established equality of topological entropy and inflation word entropy, relying on ad hoc methods. We present a unified scheme, proving that inflation word entropy and topological entropy in fact coincide. The topological entropy is approximated by a converging series of upper and lower bounds which, in many cases, lead to an analytic expression.
Highlights
Random substitution systems provide a model for structures that exhibit both longrange correlations and a positive topological entropy
We show that for a large class of random substitutions, that we will call semi-compatible, the inflation word entropy reproduces the value of the topological entropy
Definition 5 The language of a primitive random substitution θ is given by all words that appear as a subword of some inflation word
Summary
Random substitution systems provide a model for structures that exhibit both longrange correlations and a positive topological entropy. The nontrivial topological entropy is a property that distinguishes random substitutions from usual (deterministic) substitutions which are known to have a complexity function that increases at most linearly in the primitive case [3]. It has been the subject of recent. We present a way to calculate the inflation word entropy efficiently, yielding a closed form expression in many cases This result reproduces all of the known values of topological entropy mentioned above, providing much simpler proofs in some of the cases.
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