Mixing Subshift Of Finite Type Research Articles

Lyapunov irregular set of Banach cocycles

Let be a topologically mixing subshift of finite type and let be a Hölder continuous map where is the set of invertible elements of a Banach ring B. We prove that either all ergodic measures have the same maximal Lyapunov exponent, or the irregular set is residual in and has full upper capacity topological entropy of . We also prove for generic fiber-bunched cocycles, similar statements can be strengthened by considering the Bowen topological entropy.

On von Neumann regularity of cellular automata

We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and Törmä that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage of the same period. We show that the non-regular ECA 9 and 28 cannot be proven non-regular using these methods. We also show that a random cellular automaton is non-regular with high probability.

Characteristic measures for language stable subshifts

We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. It follows readily from a classical result of Parry that the full shift on finitely many symbols, and more generally any mixing subshift of finite type, supports such a measure. Frisch and Tamuz recently dubbed such measures characteristic, and further showed that every zero entropy subshift has a characteristic measure. While it remains open if every subshift over a finite alphabet has a characteristic measure, we define a new class of shifts, which we call language stable subshifts, and show that these shifts have characteristic measures. This is a large class that is generic in several senses and contains numerous positive entropy examples.

A topological dynamical system with two different positive sofic entropies

A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. This paper exhibits an explicit example of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-colorings of random hyper-graphs.

Quantitative recurrence properties in the historic set for symbolic systems

Let be a topologically mixing subshift of finite type on an alphabet consisting of m symbols. Let be a continuous function. For a positive function and a continuous positive function , define the sets of recurrence points and In this paper, we give quantitative estimates of the sets of recurrence points, that is, can be expressed by ψ and is the solution of the Bowen equation of topological pressure.

Boundedness, compactness, and invariant norms for Banach cocycles over hyperbolic systems

We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle A \mathcal {A} takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We consider the periodic data of A \mathcal {A} , i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of A \mathcal {A} is uniformly quasiconformal or bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Hölder continuous family of norms. We also obtain a general result on existence of a measurable family of norms invariant under a cocycle.

Uncountably many maximizing measures for a dense subset of continuous functions

Ergodic optimization aims to single out dynamically invariant Borel probability measures which maximize the integral of a given ‘performance’ function. For a continuous self-map of a compact metric space and a dense set of continuous functions, we show the existence of uncountably many ergodic maximizing measures. We also show that, for a topologically mixing subshift of finite type and a dense set of continuous functions there exist uncountably many ergodic maximizing measures with full support and positive entropy.

Nonconventional polynomial CLT

We obtain a functional central limit theorem (CLT) for sums of the form , where is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, is a certain centralizing constant and are arbitrary polynomials taking on positive integer values on positive integers, i.e. polynomials satisfying , where is the set of natural numbers. For polynomial ’s this CLT generalizes [The Annals of Probability, 2014, Vol. 42, No.2, 649–688] which allows only linear ’s to have the same polynomial degree. We also prove that exists and provide necessary and sufficient conditions for its positivity, which is equivalent to the statement that the weak limit of is not zero almost surely. Finally, we study independence properties of the increments of the limiting process. Our proofs require studying asymptotic densities of special subsets of , which is done in a separate section. As in [The Annals of Probability, 2014, Vol. 42, No.2, 649-688], our results hold true when , where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when , where is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.

The automorphism group of a minimal shift of stretched exponential growth

The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial growth, we further show that any finitely generated, torsion free subgroup of Aut(X) is virtually nilpotent.

Nonconventional limit theorems in discrete and continuous time via martingales

We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_1(n)),\ldots, X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced by an integral. Here $X(n),n\geq0$ is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, $F$ is a continuous function with polynomial growth and certain regularity properties, $\bar{F}=\int F\,d(\mu\times\cdots\times\mu)$, $\mu$ is the distribution of $X(0)$ and $q_i(n)=in$ for $i\le k\leq\ell$ while for $i>k$ they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when $q_i$'s are polynomials of increasing degrees. These results decisively generalize [Probab. Theory Related Fields 148 (2010) 71-106], whose method was only applicable to the case $k=2$ under substantially more restrictive moment and mixing conditions and which could not be extended to convergence of processes and to the corresponding continuous time case. As in [Probab. Theory Related Fields 148 (2010) 71-106], our results hold true when $X_i(n)=T^nf_i$, where $T$ is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when $X_i(n)=f_i({\Upsilon }_n)$, where ${\Upsilon }_n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure. Moreover, our relaxed mixing conditions yield applications to other types of dynamical systems and Markov processes, for instance, where a spectral gap can be established. The continuous time version holds true when, for instance, $X_i(t)=f_i(\xi_t)$, where $\xi_t$ is a nondegenerate continuous time Markov chain with a finite state space or a nondegenerate diffusion on a compact manifold. A partial motivation for such limit theorems is due to a series of papers dealing with nonconventional ergodic averages.

Strong mixing subshift of finite type and Hausdorff measure of its chaotic set

We present a sufficient and necessary condition for the subshift of finite type to be a measure-preserving transformation or to be a strong mixing measure-preserving transformation with respect to the Hausdorff measure. It is proved that a strong mixing subshift of finite type has a chaotic set with full Hausdorff measure.

Nonconventional limit theorems

The polynomial ergodic theorem (PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337–349, 1987) and attracted substantial attention in ergodic theory studies the limits of expressions having the form $${1/N\sum_{n=1}^NT^{q_1(n)}f_1 \cdots T^{q_\ell (n)} f_\ell}$$ where T is a weakly mixing measure preserving transformation, f i ’s are bounded measurable functions and q i ’s are polynomials taking on integer values on the integers. Motivated partially by this result we obtain a central limit theorem for even more general expressions of the form $${1/\sqrt{N}\sum_{n=1}^N (F(X_0(n),X_1(q_1(n)),X_2(q_2(n)), \ldots, X_\ell(q_\ell(n)))-\overline F)}$$ where X i ’s are exponentially fast ψ-mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, $${\overline F=\int Fd(\mu_0\times\mu_1\times \cdots\times\mu_\ell)}$$ , μ j is the distribution of X j (0), and q i ’s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q i ’s are polynomials of growing degrees. When F(x 0, x 1, . . . , x l ) = x 0 x 1 x 2 . . . x l exponentially fast α-mixing already suffices. This result can be applied in the case when X i (n) = T n f i where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as in the case when X i (n) = f i (ξ n ) where ξ n is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.

Recurrence in pairs

AbstractWe introduce and study the notion of weak product recurrence. Two sufficient conditions for this type of recurrence are established. We deduce that any point with a dense orbit in either the full one-sided shift on a finite number of symbols or a mixing subshift of finite type is weakly product recurrent. This observation implies that distality does not follow from weak product recurrence. We have therefore answered, in the negative, a question posed by Auslander and Furstenberg.

Singular perturbation of symbolic dynamics via thermodynamic formalism

AbstractWe consider singular perturbation of a mixing subshift of finite type by means of thermodynamic formalism. In our formulation, the perturbed systems are described by a family of potentials {Φ(α,⋅)} with large parameter α on a fixed subshift of finite type, and the original (unperturbed) system is characterized as the system at infinity obtained by collapsing the perturbed system upon taking $\alpha \to \infty $. We apply our formulation to the collapse of cookie-cutter systems and dispersing open billiards.

Morphisms from non-periodic \mathbb{Z}^{2} subshifts I: constructing embeddings from homomorphisms

In this paper we introduce foundational techniques and prove the following: if X is a \mathbb{Z}^d subshift without periodic points, if Y is a \mathbb{Z}^d square mixing subshift of finite type containing a finite orbit and if there exists a homomorphism X\rightarrow Y, then X embeds into Y if and only if h(X)<h(Y). For the proof, clopen markers are used to generate Voronoi tiles whose thickened boundaries are coded using the homomorphism. The entropy gap and the square mixing permit the construction of an injective code on the tile interiors. A second paper will show that \mathbb{Z}^2 square filling mixing shifts of finite type are square mixing and that homomorphisms exist, resulting in an extension of Krieger's Embedding Theorem to \mathbb{Z}^2 subshifts.

RECURRENCE, DIMENSION AND ENTROPY

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

Directional entropy of rotation sets

For (X,T) a mixing subshift of finite type, and f:X→ R d of summable variation, we give a characterisation of directional entropy ( see [4]) at interior and exposed points of the corresponding generalised rotation set.

Rotation, entropy, and equilibrium states

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.

Regularity of solutions to the measurable Livsic equation

In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of: (a) 0-transformations; (b) rational maps; and (c) planar maps with indifferent periodic points. Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity. 0. INTRODUCTION In 1972 Livsic showed that given a mixing subshift of finite type T: X -* X and a H6lder continuous function c: X -* IR then any solution to the equation (1) c(x) u(Tx) u(x) with u: X ---1iR measurable and essentially bounded (with respect to an equilibrium measure for a H6lder continuous function) must have a continuous version i.e. eu': X -* R continuous such that u(x) = u'(x) a.e. [2], [3] This is an elementary type of rigidity result. Livsic also applied this result to Anosov systems. Another approach to this result for subshifts of finite type and using Perron-Frobenius type operators appeared in [6]. In this note we shall discuss some simple generalisations of Livsic's result to other examples of dynamical systems. We shall be interested in systems with discontinuities or which are not uniformly hyperbolic and thus are not covered by Livsic's original results. In this context the Perron-Frobenius type operator method adapts easily to prove the generalisations of Livsic's original results. To illustrate this approach we shall apply this method to 3-transformations (section 1), rational maps (section 2) and certain multi-dimensional maps (section 3). 1. }5-TRANSFORMATION We begin by recalling the definition of the well-known 3-transformation on the unit interval. Let / > 1 and define T: [0,1) -* [0,1) by T(x) = fx (mod 1). Let v denote Lebesgue measure on the interval [0,1] and Ll ([0, 1], 13, v) denotes the space of integrable functions on [0,1]. Received by the editors August 5, 1996. 1991 Mathematics Subject Classification. Primary 58Fxx. (?)1999 American Mathematical Society 559 This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms 560 M. POLLICOTT AND M. YURI Definition. We define a Perron-Frobenius operator Li : Ll ([O, 1], 6,B v) --Ll ([O, 1], 6,B v) by Ty= 13 where the summation is over the [,3] or [a1] + 1 pre-images of x. We denote by BBV the space of functions of bounded variation i.e. if f E BBV then we have that var(f) :=sup{ZEif(Xi+1)-f(xi)I: O xoX 0 for x E [0,1). We shall also need the following more general operator. Definition. Given a Lipschitz function c: [0,11 --? R. We define a c-weighted Perron-Frobenius operator L2: L' ([O, 1], B, v) -* L1 ([O, 1], 13, v) by L2 f (x) = E ecp f (y) TY=X (When c= 0 then L2 clearly reduces to Lp.) It is convenient to denote cn (x) = c(x) + c(Tx) +.. + c(Tn lx) for n > 1. The following result is important in proving Theorem 1. Lemma 2. (i) Vf E L1([0, 1], B, v) we have I2nf h(f fdv) in the L1 norm, (ii) Let c: [0,11 -? IR be a Lipschitz function. There exists w E BBV with w > 0 and A > 0 such that the sequence of functions An (T-y=x BBV. The convergence Ijf -* h(J f dv), as n -4 +oo is proved in [9, Theorem 10] when f is uniformly continuous. Since any function in L1 ([O,1], B, v) can be approximated from above and below by such functions, this is sufficient. This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms REGULARITY OF SOLUTIONS l'O THE MEASURABLE LIVSIC EQUATION 561 (ii) The operator L2: BBV -BBV has a simple maximal eigenvalue A, say, with positive eigenvector w E BBV ([9, Lemma 9]). Moreover, there exists a measure ,t on [0, 1] such that fn =~ --> w f dti, as n +oo [9, Theorem 10]. 1] Our main result on the measurable Livsic equation for fl-transformations is the following. Theorem 1. If c: [0, 1t -* IR is Lipschitz then for any measurable and essentially bounded solution u to (1') c(x) = u(Tx) u(x) there exists uo(x) = v(x) log h(x), where v(x) E BBV such that u == uo a.e. (nt). Proof. We can make the choice f 1 and then iterating the operator 12 introduced above we can write (L n1)(X)= ASn fn (X) Tny3/X In particular, substituting from the identity (1') we get that f Ix eCn(y) fn (x) An E , n

Automorphisms of the dimension group and gyration numbers

O. Introduction 1. The sign-gyration-compatibility-condition 2. Gyration and orbit-sign numbers 3. Construction of sgcc A m 4. Application of sgcc A ~ SUMMARY Let Aut(OA) denote the group of automorphisms of a subshift of finite type (XA'OA) built from a primitive matrix A. We show that the sign-gyrationcompatibility-condition homomorphism SGCC A, m defined on Aut( 0 A) factors through the group Aut(s A) of automorphisms of the dimension group. This is used to find a mixing subshift of finite type with a permutation of fixed points that cannot be lifted to an automorphism of the shift. We also give an example of a mixing subshift of finite type where the dimension group representation is not surjective. This example is used in [KR3] to give examples of subshifts of finite type (reducible, with two mixing components) that are shift equivalent but not strong shift equivalent over the nonnegative integers. O. INTRODUCTION