On winning shifts of marked uniform substitutions

The second author introduced with I. T\"orm\"a a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has a predetermined (possibly finite) choice sequence $\alpha_1$, $\alpha_2$, $\ldots$ of integers such that on round $n$ the player $A$ chooses a subset $S_n$ of size $\alpha_n$ of some fixed finite alphabet and the player $B$ picks a letter from the set $S_n$. The outcome is determined by whether the word obtained by concatenating the letters $B$ picked lies in a prescribed target set $X$ (a win for player $A$) or not (a win for player $B$). Typically, we consider $X$ to be a subshift. The winning shift $W(X)$ of a subshift $X$ is defined as the set of choice sequences for which $A$ has a winning strategy when the target set is the language of $X$. The winning shift $W(X)$ mirrors some properties of $X$. For instance, $W(X)$ and $X$ have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that $W(X)$ and $X$ have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.Comment: Extended version of a paper presented at RuFiDiM I

The Analyticity of a Generalized Ruelle's Operator

In this work we propose a generalization of the concept of Ruelle operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle operator, that generalizes both the Ruelle operator proposed in [BCLMS] and the Perron Frobenius operator defined in [Bowen]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle operator and present some examples

KMS States, Entropy and the Variational Principle in full C*-dynamical systems

To any periodic, unital and full C*-dynamical system (A, \alpha, R) an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron-Frobenius type theorem asserts the existence of KMS states at inverse temperatures equal the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system and the relationship between their topological entropy and inverse temperatures of extremal KMS states are given. Noncommutative shift maps are considered. It is shown that their entropy is bounded by the sum of the entropy of the associated subshift and a suitable entropy computed in the homogeneous subalgebra. Examples are discussed among Matsumoto algebras associated to certain non finite type subshifts. The CNT entropy is compared to the classical measure-theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy of subshifts is obtained for the noncommutative shift of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz-Krieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done.Comment: 52 pages, AMSTeX. An error in Prop. 7.3 v1 has been corrected, and related text in sections 7-9 has been modified. References added. Abstract modifie

Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples

In this paper we construct spectral triples $(A,H,D)$ on the symbolic space when the alphabet is finite. We describe some new results for the associated Dixmier trace representations for Gibbs probabilities (for potentials with less regularity than H\"older) and for a certain class of functions. The Dixmier trace representation can be expressed as the limit of a certain zeta function obtained from high order iterations of the Ruelle operator. Among other things we consider a class of examples where we can exhibit the explicit expression for the zeta function. We are also able to apply our reasoning for some parameters of the Dyson model (a potential on the symbolic space $\{-1,1\}^\mathbb{N}$) and for a certain class of observables. Nice results by R. Sharp, M.~Kesseb\"ohmer and T.~Samuel for Dixmier trace representations of Gibbs probabilities considered the case where the potential is of H\"older class. We also analyze a particular case of a pathological continuous potential where the Dixmier trace representation - via the associated zeta function - is not true.Comment: the tile was modified and there are two more author

Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow \Gamma$ and vector towers $\overleftarrow v$ on $\overleftarrow \Gamma$, which allows us to efficiently describe all invariant measures $\mu = \mu^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $\varphi: F_N \to F_N$ is determined, when $\varphi$ is represented by a train track map.Comment: 52 pages, 3 figures. This is rather a new paper than a new version of the old one. The setting is much more general, and also closer to a symbolic dynamics spirit. Also, some of the work from the original paper has been removed and will be taken up in a forthcoming paper. Accepted in Journal of London Mathematical society. To appear 201