That's the only proof that I know of for the existence of fully supported, ergodic, zero entropy measure on $\Sigma$. We also prove that if G is countable and X V G is a strongly irreducible linear subshift, then every injective linear cellular automaton : X X is surjective. Such measures actually form a residual subset of the space of all invariant probability measures on $\Sigma$, for which see "ergodic theory on compact spaces" by Denker & Grillenberger & Sigmund. if X V G is a strongly irreducible linear subshift of nite type and : X X is a linear cellular automaton, then is surjective if and only if it is pre-injective. So you only need to construct an example of a fully supported, ergodic measure of zero entropy. Yet another way of producing a counter-example is to look inside the class of zero entropy processes because Markov measures always have positive entropy and if IIRC, this is also true for Gibbs measures. We illustrate this correspondence by using the Potts model and a model of our own, inspired by the vertex models in statistical mechanics.
Irreducible subshift code#
In fact, any time you have two different ergodic measures $\mu, \mu'$ on an mixing SFT projecting to the same measure $\pi\mu = \pi\mu'$ via some finite-to-one factor code onto a mixing SFT, the image $\pi\mu$ cannot be a Gibbs state in the sense of "R W"'s answer, let alone a Markov measure. between a simplex of measures of maximal entropy of a strongly irreducible subshift of nite type and a simplex of equilibrium states of the corresponding statistical mechanics model. It is probably one of the simplest counter-examples for your question.
Irreducible subshift full#
Following this terminology, we can say that the image measure $\pi\mu_p$, for $0 < p < 1/2$, in Example 2.9 in the preprint is a sofic, non-Markov measure on the full two-shift. (SFT) of positive entropy are isomorphic but they have some order structure. For a subshift Z AG, we say Z is strongly irreducible iff iZ is. The unordered fundamental groups of all irreducible subshifts of finite type.
In the preprint, as in many related literature, "Markov measure" is defined to be any invariant probability measure on any irreducible SFT (subshift of finite type) with full support and with n-step Markov property for some n, and "sofic measure" is defined to be images of such measures under factor codes. Some basic facts about strongly irreducible subshifts. I'll refer to the arxiv v2 of the preprint. Another strict generalization I'd like to mention is (stationary) hidden Markov chains, for which see "Hidden Markov processes in the context of symbolic dynamics" by Boyle & Petersen. Associated with these tilings there is a natural subshift of finite type, which is shown to be irreducible. It feels like that question should have an easy answer, but somehow I don't get it.Īs "R W"s answer points out, Gibbs measures (of regular enough potentials, such as Holder continuous potentials) are a strict generalization of Markov measures. The apartments of \cB are tiled by triangles, labelled according to \Gamma-orbits. However, in the described set-up I could not find one. It is quite easy to construct counter-examples if I drop certain assumptions. The class of normal subshifts includes irreducible nontrivial topological Markov shifts, irreducible nontrivial sofic shifts, synchronized systems. Let $\Omega = \$ are the elements of a compatible stochastic matrix and $p$ is its unique unity stochastic eigenvector? It seems rather intuitive, however, I was not able to proof it yet: Michael S.I have this question I have been struggling with for a while.Natasha Jonoska, Subshifts of Finite Type, Sofic Systems and Graphs, (2000).
Theorem 8.1.16 under the assumption of strong irreducibility. Substitutions in dynamics, arithmetics and combinatorics. a subshift of X, is conjugate to a shift of finite type in which every symbol can be. Berthé, Valérie Ferenczi, Sébastien Mauduit, Christian Siegel, A. David Damanik, Strictly Ergodic Subshifts and Associated Operators, (2005).Introduction to Dynamical Systems (2nd ed.). Matthew Nicol and Karl Petersen, (2009) " Ergodic Theory: Basic Examples and Constructions",Įncyclopedia of Complexity and Systems Science, Springer
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