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We consider [Formula: see text]-mixing dynamical systems [Formula: see text] and we find conditions on families of sets [Formula: see text] so that [Formula: see text] tends in law to an exponential random variable, where [Formula: see text] is the entry time to [Formula: see text]

Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that K can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of K as well as the set of elements in K with unique codings using the machinery of Mauldin and Williams (1988 Trans. Am. Math. Soc. 309 811–29). We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of K with multiple codings. Secondly, in the setting of β-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. Motivated by this application, we prove that the set of all the unique codings is a subshift of finite type if and only if it is a sofic shift. This equivalent condition was not mentioned by de Vries and Komornik (2009 Adv. Math. 221 390–427, theorem 1.8). Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the survivor set can be identified with a subshift of finite type. The third application partially answers a problem posed by Alcaraz Barrera (2014 PhD Thesis University of Manchester).

Let $K\subseteq\mathbb{R}$ be the unique attractor of an iterated function system. We consider the case where $K$ is an interval and study those elements of $K$ with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence of this, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams \cite{MW}. The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of \cite{DKK}, \cite{K1}, \cite{K2}, and \cite{MK}., 17 pages, 1 figure. We have changed some vague places and inappropriate citations

In this work we consider an ensemble of random $\mathbb{Z}^d$-shifts of finite type ($\mathbb{Z}^d$-SFTs) and prove several results concerning the behavior of typical systems with respect to emptiness, entropy, and periodic points. These results generalize statements made in [26] regarding the case $d=1$. &nbsp Let $\mathcal{A}$ be a finite set, and let $d \geq 1$. For $n$ in $\mathbb{N}$ and $\alpha$ in $[0,1]$, define a random subset $\omega$ of $\mathcal{A}^{[1,n]^d}$ by independently including each pattern in $\mathcal{A}^{[1,n]^d}$ with probability $\alpha$. Let $X_{\omega}$ be the (random) $\mathbb{Z}^d$-SFT built from the set $\omega$. For each $\alpha \in [0,1]$ and $n$ tending to infinity, we compute the limit of the probability that $X_{\omega}$ is empty, as well as the limiting distribution of entropy of $X_{\omega}$. Furthermore, we show that the probability of obtaining a nonempty system without periodic points tends to zero. &nbsp For $d>1$, the class of $\mathbb{Z}^d$-SFTs is known to contain strikingly different behavior than is possible within the class of $\mathbb{Z}$-SFTs. Nonetheless, the results of this work suggest a new heuristic: typical $\mathbb{Z}^d$-SFTs have similar properties to their $\mathbb{Z}$-SFT counterparts.

We introduce a method of analysing longitudinal data in n≥1 variables and a population of K≥1 observations. Longitudinal data of each observation is exactly coded to an orbit in a two-dimensional state space Sn. At each time, information of each observation is coded to a point (x,y)∈Sn, where x is the physical condition of the observation and y is an ordering of variables. Orbit of each observation in Sn is described by a map that dynamically rearranges order of variables at each time step, eventually placing the most stable, least frequently changing variable to the left and the most frequently changing variable to the right. By this operation, we are able to extract dynamics from data and visualise the orbit of each observation. In addition, clustering of data in the stable variables is revealed. All possible paths that any observation can take in Sn are given by a subshift of finite type (SFT). We discuss mathematical properties of the transition matrix associated to this SFT. Dynamics of the population is a nonautonomous multivalued map equivalent to a nonstationary SFT. We illustrate the method using a longitudinal data of a population of households from Agincourt, South Africa.

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Holder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map \([\cdot] :G \to G/[G,G]\). This is obtained via an analysis involving subshifts of finite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus \(\geq 2\) is also discussed.

We investigate factor maps of higher-dimensional subshifts of finite type. In particular, we are interested in how the number of ergodic measures of maximal entropy behaves under such factor maps. We show that this numberis preserved under almost invertible maps, but not in general under finite to one factor maps. One of our tools, which is of independent interest, is a higher-dimensional characterization of entropy-preserving factor maps that extends the well-known one-dimensional characterization result.

Let ([sum ]A, T) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Φ:[sum ]A → Rd be a continuous function. Denote by σΦ(x) the ergodic limit limn→∞n−1 [sum ]n−1j=0 Φ(Tjx) when the limit exists. Possible ergodic limits are just mean values ∫ Φdμ for all T-invariant measures. For any possible ergodic limit α, the following variational formula is proved:[formula here]where hμ denotes the entropy of μ and htop denotes topological entropy. It is also proved that unless all points have the same ergodic limit, then the set of points whose ergodic limit does not exist has the same topological entropy as the whole space [sum ]A

We present a method to compute rigorous upper bounds for the topological entropy h(T , A) of a continuous map T with respect to a fixed (coarse) partition of the phase space A. Long trajectories are not used; rather a single application of T to the phase space produces a topological Markov chain which contains all orbits of T , plus some additional spurious orbits. By considering the Markov chain as a directed graph, and labelling the arcs according to the fixed partition, one constructs a sofic shift with topological entropy greater than or equal to h(T , A). To exactly compute the entropy of the sofic shift, we produce a subshift of finite type with equal entropy via a standard technique; the exact entropy calculation for subshifts is then straightforward. We prove that the upper bounds converge monotonically to h(T , A) as the topological Markov chains become increasingly accurate. The entire procedure is completely automatic. © 2001 Elsevier Science B.V. All rights reserved. MSC: 37B40; 94A17; 37M25

For a dynamical system (X, T) and function f : X Rd we consider the corresponding generalised rotation set. This is the convex subset of Rd consisting of all integrals of f with respect to T-invariant probability measures. We study the entropy H(e) of rotation vectors e, and relate this to the directional entropy lHl(L) of Geller & Misiurewicz. For (X, T) a mixing subshift of finite type, and f of summable variation, we prove that if the rotation set is strictly convex then the functions -H and H are in fact one and the same. For those rotation sets which are not strictly convex we prove that 1H((e) and H(e) can differ only at non-exposed boundary points e.

In this paper, two recursive formulas for computing the spatial entropy of two-dimensional subshifts of finite type are given. The exact entropy of a nontrivial example arising in cellular neural networks is obtained by using such formulas. We also establish some general theory concerning the spatial entropy of two-dimensional subshifts of finite type. In particular, we show that if either of the transition matrices is rank-one, then the associated exact entropy can be explicitly obtained. The generalization of our results to higher dimension can be similarly obtained. Furthermore, these formulas can be used numerically for estimating the spatial entropy.

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.

We establish the multifractal analysis of hyperbolic flows and of suspension flows over subshifts of finite type. A non-trivial consequence of our results is that for every Holder continuous function non-cohomologous to a constant, the set of points without Birkhoff average has full topological entropy.

We continue the study of mixing properties of generic hyperbolic flows started in an earlier paper (D. Dolgopyat. Prevalence of rapid mixing in hyperbolic flows. Erg. Th.& Dyn. Sys.18 (1998), 1097–1114). Our main result is that generic suspension flow over subshifts of finite type is exponentially mixing. This is a quantitative version of an earlier result of Parry and Pollicott (W. Parry and M. Pollicott. Stability of mixing for toral extensions of hyperbolic systems. Proc. Steklov Inst.216 (1997), 354–363).

We show that every shift of finite type $X$ with positive entropy has proper subshifts of finite type with entropy strictly smaller than the entropy of $X$, but with entropy arbitrarily close to the entropy of $X$. Consequently, $X$ contains an infinite chain of subshifts of finite type which is strictly decreasing in entropy.

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

We prove large-deviation results for empirical measures, with respect to an arbitrary g -measure on a subshift of finite type, by using an ergodic theorem for relative entropy we establish. Extending some results of Cajar on Billingsley dimensions to g -measures, we relate the relative entropy to the dimensions of saturated sets. This allows a dimensional interpretation of the large-deviation results and the study of the set of non-generic points which are proved to be of full dimension.

The pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism. The present paper gives a more concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function Φ to its long-term time averages through the Hausdorff and packing dimensions of the subsets on which Φ has prescribed long-term time-average values. Functions Φ with values in ℝd are considered. For those Φ depending only on finitely many symbols, we get complete results, unifying and completing many partial results.

We introduce notions of expansiveness, conjugation, and specification for random bundle transformations and derive the uniqueness of equilibrium states for a large class of functions. We consider both invertible and noninvertible cases and discuss the results in the random subshifts case. As an example of such systems we introduce random sofic shifts which can be described both via random graphs and as factors of random subshifts of finite type. Based on the random graph description we discuss large deviation results for random sofic shifts.

We study the natural symbolic dynamics associated with piecewise continuous, non-invertible, dynamical systems. Our study is centered primarily on the relationship between the point-set topological properties of the partition of the system and the symbolic coding. We prove that for a class of maps locally preserving distances with regular partition, the associated symbolic dynamics cannot embed subshifts of finite type of positive entropy. Hence, in particular, almost sofic subshifts obtained from the symbolic dynamics have zero entropy. However, there are examples in Euclidean spaces of systems with non-regular partitions for which the coding maps can be surjective, particularly embedding all subshifts. For all such examples, the associated group of isometries is a subgroup of $O(\mathbb{R}, N)$.

For a given mapf:X→Xf: X \to Xand an observableφ:X→Rd,\varphi : X \to \mathbb {R} ^{d},rotation vectors are the limits of ergodic averages ofφ.\varphi .We study which part of the topological entropy offfis associated to a given rotation vector and which part is associated with many rotation vectors. According to this distinction, we introduce directional and lost entropies. We discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps.

We provide necessary and sufficient conditions for a suspension flow, over a subshift of finite type, to mix faster than any power of time. Then we show that these conditions are satisfied if the flow has two periodic orbits such that the ratio of the periods cannot be well approximated by rationals.

We introduce a notion of generalized equilibrium states which include Riesz products G-measures and classical equilibrium states on a subshift of finite type. We prove a rather precise estimate for ∥Pnƒ∥∞ where Pn are the average operators (see Introduction). As a consequence we obtain an estimate for ∥Lgnƒ¦∞ where Lg is a transfer operator on a subshift of finite type. A second consequence concerns the almost everywhere convergence of some lacunary series; the obtained result is new in the case of G-measures studied by G. Brown and A.H. Dooley and improves to some extend a result of J. Peyriere in the case of Riesz products.

On a subshift of finite type (SFT) we introduce a pseudometric d given by a nonnegative matrix B satisfying the cycle condition. We show that the Hausdorff dimension of this SFT with respect to d is given by the Mauldin-Williams formula. If the ratio of the logarithms of any two nonzero entries of B is rational, we show that this Hausdorff dimension can be expressed essentially in terms of the logarithm of the spectral radius of a certain digraph. We apply our results to the Hausdorff dimension of the limit set of finitely generated free groups of isometrics of infinite trees. To each finitely generated subgroup G of a given finitely generated free group F , we attach an invariant p ( G ), which gives the rate of growth of all words G of length l at most with respect to a fixed set of minimal generators of F . We show that p ( G ) is the spectral radius of a digraph Δ( G ) induced by G . Then H ⩽ G ⩽ F ⇏ p ( G ) ⩾ p ( H ). Moreover, p ( G ) = p ( H ) ⇔ [ G : H ]

In this paper we study small, random, diffeomorphism-type perturbations of an Axiom A basic set. By means of the structural stability of such a basic set with respect to time-dependent perturbations and by means of the Markov partition of the basic set, we apply the thermodynamic formalism of random subshifts of finite type to this situation, obtaining some ergodic-theoretic results concerning equilibrium states.

We present a special model which is a caricature of the collinear three-body problem. Near triple-collision behavior for the model is governed by the collision manifold. Using the dynamics on the collision manifold, we study the allowed sequences of bounces by generating a partition on a Poincare slice in the flow. When the ratio of masses is small, we use two subshifts of finite type to describe which sequences of bounces are guaranteed and which are excluded. There is a third class of sequences which are neither guaranteed nor excluded. We also describe the bifurcations of this system as the ratio of masses approaches zero.

Let X 0 X_{0} , X X be mixing one-sided subshifts of finite type such that X 0 ⊆ X X_{0}\subseteq X . We show a necessary and sufficient condition for the existence of mixing Markov shifts Y 0 Y_{0} , Y Y , Y 0 ⊆ Y Y_{0}\subseteq Y , and a conjugacy π : Y → X \pi : Y\to X with π ( Y 0 ) = X 0 \pi (Y_{0})=X_{0} , such that the sets of letters appearing in both systems are the same, more precisely, L 1 ( Y 0 ) = L 1 ( Y ) L_{1}(Y_{0})=L_{1}(Y) .

In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a tnixture of ideas from both "thermodynamic" ergodic theory and the automaton associated to strongly Markov groups. 0. INTRODUCTION In [5] Cannon showed that for fundamental groups of many manifolds of negative curvature (including, for example, compact manifolds) the generating function for word length is a rational function. To explain the implications of this property, we recall that the growth of the quantity N(n) = Card{g fE r Igl = n} -was studied by Milnor in his fundamental paper [13], where he obtained estimates using comparison with the growth of volume in the universal cover. Cannon used purely combinatorial methods to show that the generating function is rational. In particular, this implies that we can find constants yBi and Ci, and positive integers ki (i = :1, . . ., N) such that N(n) = #1_1 Cinkitn. The key step in Cannon's approach was to associate to the grol1p an automaton. In this article we shall augment this with the notion of "weighting". This allows us to draw upon the well-known theory of Thermodynamic Formalism to prove a number of new results which can be viewed as weighted analogues of the above results. It has long been understood that this approach must havew close connections with symbolic dynamics, Markov partitions, subshifts of finite type (cf. [1], [20],[21] and, in particular, the work of Bourdon [4]) and by extension to the whole paraphernalia of the area of dynamical systems collectively called "Thermodynamic Formalism". In this article, we shall elaborate on this connection as an integral part of our analysis. Let TEln, for some n > 2, denote n-dimensional hyperbolic space (i.e. the unique n-dimensional simply connected, complete Riemannian manifold wi-th all sectional Received by the editors May 23, 1995. 1991 Mathematics Subject Classification. Primary 20F32, 22E40, 58E40; Secondary 11F72, 20F10, 58F20.

Follow this and additional works at: http://opensiuc.lib.siu.edu/math_articles Part of the Mathematics Commons This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems following peer review. The definitive publisher-authenticated version "Invariants of twist-wise flow equivalence," Discrete and Continuous Dynamical Systems, 4(3), 475 484 is available online at: http://aimsciences.org/journals/dcdsA/online.jsp

C*-algebras generalizing Cuntz-Krieger algebras can be associated to hyperbolic homeomorphisms of compact metric spaces. They satisfy a non-commutative form of Spanier-Whitehead duality with respect to K-theory. We prove this for the case of subshifts of finite type. The special feature of the present situation is that the constructions are all done on the full Fock space and are very explicit, while the general theorem requires much more abstract machinery., 23 pages, Latex file

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

We associate to a Turing machine two dynamical systems which we call Turing machine with moving tape (TMT) and Turing machine with moving head (TMH). TMT are equivalent to generalized shifts of Moore (1990) and they include two-sided full shifts. TMH are shift-commuting maps of two-sided sofic systems. In both classes we characterize systems with the shadowing property, show that a bijective expansive TMT is conjugate to a subshift of finite type and that topological entropy of every TMH is zero. We conjecture that every TMT has a periodic point.

We examine a family of maps, which are motivated by a specific flow model for a manufacturing system, that act on polygons inscribed within an equilateral triangle. The family of maps is interesting because it provides an example of chaotic behaviour in one extreme and periodic behaviour in the other, with a non-standard transition between these extremes. We characterize the behaviour of this family of maps and describe the interesting phenomena which occur between the two extremes of chaos and periodicity. We also use symbolic dynamics to relate one of the maps to a particular subshift of finite type.

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures—a version of de Finetti’s theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, ‘canonical’ Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

We present constructions of countable two-dimensional subshifts of finite type (SFTs) with interesting properties. Our main focus is on properties of the topological derivatives and subpattern posets of these objects. We present a countable SFT whose iterated derivatives are maximally complex from the computational point of view, constructions of countable SFTs with high Cantor-Bendixson ranks, a countable SFT whose subpattern poset contains an infinite descending chain and a countable SFT whose subpattern poset contains all finite posets. When possible, we make these constructions deterministic, and ensure the sets of rows are very simple as one-dimensional subshifts., 38 pages, 11 figures. Extended version of arXiv:1208.2756. To appear in Fundamenta Informaticae

The aim of this short note is to compute the topological entropy for a family of skew-product maps, whose base is a subshift of finite type, and the fiber maps are homeomorphisms defined in one dimensional spaces. We show that the skew-product map does not increase the topological entropy of the subshift.

International audience; Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts of finite type (SFTs) in dimension d > 1. In particular, we prove that the factorization problem is Σ03-complete and that the conjugacy and embedding problems are Σ01-complete in the arithmetical hierarchy.

In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibers. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the same holds for a residual subset of the space of skew products., 8 pages. To appear in Nonlinearity

To an $r$-dimensional subshift of finite type satisfying certain special properties we associate a $C^*$-algebra $\cA$. This algebra is a higher rank version of a Cuntz-Krieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: if $\G$ is a group acting freely on the vertices of an $\wt A_2$ building, with finitely many orbits, and if $\Omega$ is the boundary of that building, then $C(\Om)\rtimes \G$ is the algebra associated to a certain two dimensional subshift.

In this paper we unify and extend many of the known results on the dimension of deterministic and random Cantor-like sets in ℝ n , and apply these results to study some problems in dynamical systems. In particular, we verify the Eckmann-Ruelle Conjecture for equilibrium measures for Holder continuous conformal expanding maps and conformal Axiom A# (topologically hyperbolic) homeomorphims. We also construct a Holder continuous Axiom A# homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere. this example shows that the non-conformal Holder continuous version of the Eckmann-Ruelle Conjecture is false. The Cantor-like sets we consider are defined by geometric constructions of different types. The vast majority of geometric constructions studied in the literature are generated by a finite collection ofp maps which are either contractions or similarities and are modeled by the full shift onp symbols (or at most a subshift of finite type). In this paper we consider much more general classes of geometric constructions: the placement of the basic sets at each step of the construction can be arbitrary, and they need not be disjoint. Moreover, our constructions are modeled by arbitrary symbolic dynamical systems. The importance of this is to reveal the close and nontrivial relations between the statistical mechanics (and especially the absence of phase transitions) of the symbolic dynamical system underlying the geometric construction and the dimension of its limit set. This has not been previously observed since no phase transitions can occur for subshifts of finite type. We also consider nonstationary constructions, random constructions (determined by an arbitrary ergodic stationary distribution), and combinations of the above.

We derive precise asymptotic formulae for large deviation probabilities for suspensions of subshifts of finite type. As a corollary, we give a stronger version of the Central Limit Theorem. We apply our results to transitive Anosov flows, giving a result describing fluctuations in the volume of Bowen balls and an asymptotic large deviation formula of a homological nature involving Schwartzmann’s winding cycle.

We consider a two-dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice. The configuration space of the spin system is a direct product of mixing subshifts of finite type. The potential is defined on the set of all squares in Z2 and decays exponentially with the linear size of the square. Via the polymer expansion technique we prove that for sufficiently high temperatures the limit Gibbs distribution is unique and has an exponential decay of correlations.

It has recently been shown that a strongly irreducible subshift of finite type in two or more dimensions may have more than one measure of maximal entropy. In this paper we obtain some results on when (i.e. for what kinds of subshifts of finite type) this happens, and when it does not. In particular, we show that the parameter of a certain subshift of finite type introduced by Burton and Steif has a critical value, below which we have a unique measure of maximal entropy, and above which we have non-uniqueness.

Consider a Gibbs potential on the integer lattice ind dimensions for a system with a finite state space. Suppose the interactions are translation invariant and have bounded range (the Ising and Potts models fit into this description). If the parameters of the potential are rational, we show how to construct an equivalent subshift of finite type, in the sense that there is a canonical bijection between Gibbs states for the potential and measures of maximal entropy for the subshift of finite type.

We give a new proof of the Ruelle operator theorem by using a setup inspired by the G-measure formalism due to Brown-Dooley.

Using affine Bruhat-Tits buildings, we associate certain subshifts of finite type to systems arising from the action of a Cartan subgroup of ap-adic semisimple Chevalley group on compact quotient spaces Γ\G. These are used to study the resulting dynamical systems.

It has recently been demonstrated that there are strongly irreducible subshifts of finite type with more than one measure of maximal entropy. Here we obtain a number of results concerning the uniqueness of the measure of maximal entropy. In addition, we construct for anyd≥2 andk a strongly irreducible subshift of finite type ind dimensions with exactlyk ergodic (extremal) measures of maximal entropy. Ford≥3, we construct a strongly irreducible subshift of finite type ind dimensions with a continuum of ergodic measures of maximal entropy.

We find some estimates for the derivatives of equilibrium states of subshifts of finite type. We prove the differentiability (with respect to the potential) of integrals of certain discontinuous functions for the equilibrium state of a potential.

For the Ising model with rational parameters we show how to construct a subshift of finite type that is equivalent to this Ising model, in that the translation invariant Gibbs measures for the Ising model and the measures of maximal entropy for the subshift of finite type can be identified in a natural way. This is generalized to the non-translation invariant case as well. We also show how to construct, given any H > 0, an ergodic measure of maximal entropy for a subshift of finite type and a continuous factor, such that the factor has entropy H.