Noncommutative entropy on generalized countable Markov shifts

Homoclinic bifurcations for the Hénon map

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.

Let $X$ be a one-sided subshift of finite type on a countable alphabet, and $T:X\to X$ the shift map. If $f:X\to{\mathbb R}$ is continuous, we provide conditions guaranteeing that $f$-maximizing measures exist, and are characterised by the condition that their support lies in a certain compact set.

This thesis is divided into two parts. In the first part, we consider an endomorphism \\alpha: A \\to A, with A being a commutative Banach algebra with unity, and X the spectrum of A. We ensure the existence of a unique partial dynamical system \\varphi: \\Delta \\subseteq X \\to X associated with the endomorphism. By adding the condition that A is a uniform algebra, we obtain an expression for the spectral radius of the weighted endomorphisms a\\alpha, a \\in A, which depends on the invariant or ergodic probabilities measures of the partial dynamical system (X, \\varphi). Moreover, this spectral radius will coincide with the spectral radius of the weighted shift operators aT, where \\alpha(a) = TaS for some operator S. In the second part of the project, for the shift operator \\sigma: \\Delta \\to X_A, where X_A is a generalized Markov shift space, we identify a *-endomorphism \\alpha: \\mathcal_A \\to \\mathcal_A associated with (X_A, \\sigma), where \\mathcal_A is the algebra whose spectrum is X_A. We give examples under conditions that guarantee the continuous extension of the shift operator and other examples where we show the impossibility of extending the shift operator continuously over the entire space. For the particular case of the renewal shift, we characterize every element of the algebra and provide an explicit formula to calculate r(a\\alpha) for all a \\in \\mathcal_A.

We prove that, given a uniformly locally constant potential f on a countable state Markov shift and suitable conditions which guarantee the existence of the equilibrium states μtf for all t, the measures μtf converge in the weak star topology as t tends to infinity.

Rigorous computation of topological entropy with respect to a finite partition

It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising a linear elastic material under no loads vanishes. We prove that the same result does not hold in general in the geometrically nonlinear case. This problem of mechanics establishes the purely geometrical result that the $\mathop{\mathrm{curl}}\nolimits $ of a sufficiently smooth two-dimensional rotation field cannot be a non-vanishing constant on a domain. It is classically known in continuum mechanics, stated first by the brothers Cosserat (SIAM J. Appl. Math. 25(3):483–491, 1973), that if a second order tensor field on a simply connected domain is at most a curl-free field of rotations, then the field is necessarily constant on the domain. It is shown here that, at least in dimension 2, this classical result is in fact a special case of a more general situation where the curl of the given rotation field is only known to be at most a constant.

We introduce the canonical-boundary representation and study its range. This conjugacy invariant homomorphism captures information about the symmetry of the Markov shift near its (canonical) boundary and exhibits which actions on the boundary can be realized by automorphisms.

We study the class $C_\pi$ of probability measures invariant with respect to the shift transformation on $K^\mathbb{Z}$ (where $K$ is a finite set of integers) which satisfies the Chapman-Kolmogorov equation for a given stochastic matrix $\Pi$. We construct a dense subset of measures in $C_\pi$ distinct from the Markov measure. When $\Pi$ is irreducible and aperiodic, these measures are ergodic but not weakly mixing. We show that the set of measures with infinite memory is $G_\delta$ dense in $C_\pi$ and that the Markov measure is the unique measure which maximizes the Kolmogorov-Sinai (K-S) entropy in $C_\pi$. We give examples of ergodic measures in $C_\pi$ with zero entropy.

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 Hölder continuous conformal expanding maps and conformal Axiom A# (topologically hyperbolic) homeomorphims. We also construct a Hölder 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 Hölder 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.

There are a variety of results in the literature proving forms of computability for topological entropy and pressure on subshifts. In this work, we prove two quite general results, showing that topological pressure is always computable from above given an enumeration for a forbidden list inducing the subshift, and that for strongly irreducible (SI) shifts of finite type, topological pressure is computable. Our results apply to subshifts on all finitely generated amenable groups with decidable word problem and generalize several previous results which applied only to Z d -subshifts. As corollaries, we obtain some results related to ground state energy and entropy, proving that the map sending φ to sup μ ∈ M σ ( X ) ∫ ϕ d μ is computable/computable from above when P X ( ϕ ) is, and that the map sending φ to its ground state/residual entropy is computable from above when P X ( ϕ ) is computable. We conclude by giving explicit bounds on computation time of P X ( ϕ ) in the Z d setting for SI subshift of finite types and locally constant and rational valued φ, and show that in the special case X = A Z 2 , this algorithm runs in singly exponential time.

We investigate the computability (in the sense of computable analysis) of the topological pressure P top(ϕ) on compact shift spaces X for continuous potentials ϕ:X→R . This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function (X, ϕ) ↦ P top(X, ϕ| X ) is not computable for a large set of shift spaces X and potentials ϕ. In particular, the entropy map X ↦ h top(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

Two cellular automata are strongly conjugate if there exists a shift-commuting conjugacy between them. We prove that the following two sets of pairs (F, G) of one-dimensional one-sided cellular automata over a full shift are recursively inseparable: (i) pairs where F has strictly larger topological entropy than G, and (ii) pairs that are strongly conjugate and have zero topological entropy.

There are well-known examples of dynamical systems for which the Birkhoff averages with respect to a given observable along some or all of the orbits do not converge. It has been suggested that such orbits could be classified using higher order averages. In the case of a bounded observable, we show that a classical result of G.H. Hardy implies that if the Birkhoff averages do not converge, then neither do the higher order averages. If the Birkhoff averages do not converge then we may denote by [α k ,β k ] the limit set of the k-th order averages. The sequence of intervals thus generated is nested: [α k+1,β k+1]⊂[α k ,β k ]. We can thus make a distinction among non-convergent Birkhoff averages; either: B1. is a point B ∞, or, B2. is an interval [α∞,β∞]. We give characterizations of the types B1 and B2 in terms of how slowly they oscillate and we give examples that exhibit both behaviours B1 and B2 in the context of full shifts on finite symbols and ‘Bowen's example’. For finite full shifts, we show that the set of orbits with type B2 behaviour has full topological entropy.

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.