![]() ![]() We prove that the the Fischer automaton is a topological conjugacy invariant of the underlying irreducible sofic shift. We characterize the Fischer automaton of an almost of finite type tree-shift and we design an algorithm to check whether a sofic tree-shift is almost of finite type. It is a meaningful intermediate dynamical class in between irreducible finite type tree-shifts and irreducible sofic tree-shifts. We define the notion of almost of finite type tree-shift which are sofic tree-shifts accepted by a tree automaton which is both deterministic and co-deterministic with a finite delay. We show that, contrary to shifts of infinite sequences, there is no unique reduced deterministic irreducible tree automaton accepting an irreducible sofic tree-shift, but that there is a unique synchronized one, called the Fischer automaton of the tree-shift. In other words, \(h(T)\) informs about the minimal number of symbols sufficient to encode the system "in real time" (i.e., without rescaling the time).We introduce the notion of sofic tree-shifts which corresponds to symbolic dynamical systems of infinite ranked trees accepted by finite tree automata. ĭefinitions By Adler, Konheim and McAndrewįor an open cover \(\mathcal\)). The most important characterization of topological entropy in terms of Kolmogorov-Sinai entropy, the so-called variational principle was proved around 1970 by Dinaburg, Goodman and Goodwyn. In particular, we derive an expression for the Perron eigenvectors of the associated adjacency matrix. Equivalence between the above two notions was proved by Bowen in 1971. eigenvectors of an irreducible subshift of nite type with the correlation between the forbidden words in the subshift. It uses the notion of \(\varepsilon\)-separated points. In metric spaces a different definition was introduced by Bowen in 1971 and independently Dinaburg in 1970. Then to define topological entropy for continuous maps they strictly imitated the definition of Kolmogorov-Sinai entropy of a measure preserving transformation in ergodic theory. Their idea to assign a number to an open cover to measure its size was inspired by Kolmogorov and Tihomirov (1961). ![]() The original definition was introduced by Adler, Konheim and McAndrew in 1965. Finally, we show how to extend Hedlund's results on inverses of onto endomorphisms to endomor- phisms of irreducible subshifts of finite type.
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