Tail invariant measures of the Dyck shift

We show that the one-sided Dyck shift has a unique tail invariant topologically σ-finite measure (up to scaling). This invariant measure of the one sided Dyck turns out to be a shift-invariant probability. Furthermore, it is one of the two ergodic probabilities obtaining maximal entropy. For the two sided Dyck shift we show that there are exactly three ergodic double-tail invariant probabilities. We show that the two sided Dyck has a double-tail invariant probability, which is also shift invariant, with entropy strictly less than the topological entropy. This article is a part of the author’s M.Sc. Thesis, written under the supervision of J. Aaronson, Tel-Aviv University.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

On strictly ergodic models which are not almost topologically conjugate

Answering a question raised by Glasner and Rudolph (1984) we construct uncountably many strictly ergodic topological systems which are metrically isomorphic to a given ergodic system (X, ℬ,μ, T) but not almost topologically conjugate to it. This paper is part of the second author’s Ph.D. thesis, written under the supervision of Professor A. Bellow of the Department of Mathematics, Northwestern University. The author is grateful for her encouragement and advice. We acknowledge B. Weiss for helpful comments.     

Invariant measures of minimal post-critical sets of logistic maps

We construct logistic maps whose restriction to the ω-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the ω-limit set of its critical point. Furthermore, we show that such a logistic map f can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of f for the potential −ln |f′|.     

Onσ-finite invariant measures for Markov processes

Sufficient conditions are given for the existence ofσ-finite invariant measure for conservative and ergodic Markov processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem uuder the supervision of Professor S. R. Foguel. The author wishes to thank him for his helpful advice and kind encouragement.     

Markov extensions for multi-dimensional dynamical systems

By a result of F. Hofbauer [11], piecewise monotonic maps of the interval can be identified with topological Markov chains with respect to measures with large entropy. We generalize this to arbitrary piecewise invertible dynamical systems under the following assumption: the total entropy of the system should be greater than the topological entropy of the boundary of some reasonable partition separating almost all orbits. We get a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy. We illustrate our results by quoting an application to a class of multi-dimensional, non-linear, non-expansive smooth dynamical systems. Part of this work was done at Université Paris-Sud, dép. de mathématiques, Orsay.     

Minimal models for noninvertible and not uniquely ergodic systems

Let (Y, S) be a (not necessarilly invertible) topological dynamical system on a zero-dimensional metric spaceY without periodic points. Then there exists a minimal system (X, T) with the same simplex of invariant measures as (Y, S). More precisely, there exists a Borel isomorphism between full sets inY andX such that the adjoint map on measures is a homeomorphism between the corresponding sets of invariant measures in the weak topology. As an application we construct a minimal system carrying isomorphic copies of all nonatomic invariant measures.     

Choquet simplices as spaces of invariant probability measures on post-critical sets

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak^∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the ω-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f   is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential −ln|f^′|$-\ln |f^{\prime }|$.     

On entropy of dynamical systems with almost specification

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.     

Ergodic Group Rotations, Hartman Setsand Kronecker Sequences

 Let be a homomorphism with dense image in the compact group C. If is a continuity set, i.e. its topological boundary has Haar measure 0, then is called a Hartman set. If M is aperiodic then S contains the essential information about (C, ι) or, equivalently, about the dynamical system (C, T) where T is the ergodic group rotation . Using Pontryagin’s duality the paper presents a new method to get this information from S: The set S induces a filter on which is an isomorphism invariant for (C, T) and turns out to be a complete invariant for ergodic group rotations. If one takes , , , , one gets the interesting special case of Kronecker sequences (nα) which are classical objects in number theory and diophantine analysis.     

The choquet simplex of invariant measures for minimal flows

The set of invariant measures of a compact dynamical system is well known to be a nonempty compact metrizable Choquet simplex. It is shown that all such simplices are realized already for the class of minimal flows. Moreover, sufficient is the class of 0–1 Toeplitz flows. Previously, it is proved that the set of invariant measures of the regular Toeplitz flows contains homeomorphic copies of all metric compacta.     

Embedding topological semigroups in topological groups

In [6] Rothman investigated the problem of embedding a topological semigroup in a topological group. He defined a concept calledProperty F and showed that Property F is a necessary and sufficient condition for embedding a commutative, cancellative topological semigroup in its group of quotients as an open subset. This paper announces a generalization of Rothman’s result by definingProperty E and stating that a completely regular topological semigroup S can be embedded in a topological group by a topological isomorphism if and only if S can be embedded (algebraically) in a group and S has Property E. Property E is defined by first constructing a free topological semigroup (Theorem 1.1). This construction resembles the one in [4] for a free topological group. Full details, examples, and other embedding results will appear elsewhere. Some of the results in this paper were contained in the author’s doctoral dissertation written at Rutgers University under Professor Louis F. McAuley.     

Toeplitz minimal flows which are not uniquely ergodic

Summary We study minimal symbolic dynamical systems which are orbit closures of Toeplitz sequences. We construct 0–1 subshifts of this type for which the set of ergodic invariant measures has any given finite cardinality, is countably infinite or has cardinality of the continuum.     

A loosely Bernoulli counterexample machine

In Rudolph’s paper on minimal self joinings [7] he proves that a rank one mixing transformation constructed by Ornstein [5] can be used as the building block for many ergodic theoretical counterexamples. In this paper we show that Ornstein’s transformation can be altered to create a general method for producing zero entropy, loosely Bernoulli counter-examples. This paper answers a question posed by Ornstein, Rudolph, and Weiss [6].     

Ergodic Group Rotations,Hartman Setsand Kronecker Sequences

 Let be a homomorphism with dense image in the compact group C. If is a continuity set, i.e. its topological boundary has Haar measure 0, then is called a Hartman set. If M is aperiodic then S contains the essential information about (C, ι) or, equivalently, about the dynamical system (C, T) where T is the ergodic group rotation . Using Pontryagin’s duality the paper presents a new method to get this information from S: The set S induces a filter on which is an isomorphism invariant for (C, T) and turns out to be a complete invariant for ergodic group rotations. If one takes , , , , one gets the interesting special case of Kronecker sequences (nα) which are classical objects in number theory and diophantine analysis. Received 3 November 2000; in final form 25 January 2002     

Smooth interval maps have symbolic extensions

We prove that if X denotes the interval or the circle then every transformation T:X→X of class C ^r , where r>1 is not necessarily an integer, admits a symbolic extension, i.e., every such transformation is a topological factor of a subshift over a finite alphabet. This is done using the theory of entropy structure. For such transformations we control the entropy structure by providing an upper bound, in terms of Lyapunov exponents, of local entropy in the sense of Newhouse of an ergodic measure ν near an invariant measure μ (the antarctic theorem). This bound allows us to estimate the so-called symbolic extension entropy function on invariant measures (the main theorem), and as a consequence, to estimate the topological symbolic extension entropy; i.e., a number such that there exists a symbolic extension with topological entropy arbitrarily close to that number. This last estimate coincides, in dimension 1, with a conjecture stated by Downarowicz and Newhouse [13, Conjecture 1.2]. The passage from the antarctic theorem to the main theorem is applicable to any topological dynamical system, not only to smooth interval or circle maps.     

A minimal 0–1 subshift with noncompact set of ergodic measures

Summary By a minimal 0–1 subshift we mean a pair (X, S), where S denotes the left shift on C={0, 1}^z and X is a minimal compact S-invariant subset of C. Developing some of the methods of Williams [2] of obtaining not uniquely ergodic minimal subshifts we construct such a subshift, for which the set of all ergodic measures is noncompact for the weak^* topology. In other words, the Choquet simplex of all invariant measures of the subshift is not a Bauer simplex.     

On quasi-weakly almost periodic points

The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.     

Mel’nikov-Samoilenko adiabatic stability problem

We develop a symplectic method for the investigation of invariant submanifolds of nonautonomous Hamiltonian systems and ergodic measures on them. The so-called Mel’nikov-Samoilenko problem for the case of adiabatically perturbed completely integrable oscillator-type Hamiltonian systems is studied on the basis of a new construction of “ virtual” canonical transformations. Dedicated to the memory of Viktor Koz’mich Mel’nikov, colleague and teacher, a talented Moscow mathematician, without whom the theory of dynamical systems would not be so attractive. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 787–803, June, 2006.