Compact abelian group extensions of discrete dynamical systems

Summary This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.     

Uniquely ergodic property of minimal probability measure in positive definite Lagrangian systems

Mane conjectured that every minimal measure in the generic Lagrangian systems is uniquely ergodic. In this paper, we will show that the answer to the Mane's conjecture is negative by analyzing the structure of the supports of minimal probability measures for some kinds of the Lagrangian systems.     

Fusion: a general framework for hierarchical tilings of \mathbb{R }^d

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli–Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing.     

Uniform sets for infinite measure-preserving systems

The concept of a uniform set is introduced for an ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space. The uniform sets exist inasmuch as they generate the underlying σ-algebra. This leads to the result that every ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space is isomorphic to a minimal homeomorphism on a locally compact metric space which admits a unique, up to scaling, invariant Radon measure.     

Minimality and unique ergodicity for adic transformations

We study the relationship between minimality and unique ergodicity for adic transformations. We show that three is the smallest alphabet size for a unimodular “adic counterexample”, an adic transformation which is minimal but not uniquely ergodic. We construct a specific family of counterexamples built from (3 × 3) nonnegative integer matrix sequences, while showing that no such (2 × 2) sequence is possible. We also consider (2 × 2) counterexamples without the unimodular restriction, describing two families of such maps. Though primitivity of the matrix sequence associated to the transformation implies minimality, the converse is false, as shown by a further example: an adic transformation with (2 × 2) stationary nonprimitive matrix, which is both minimal and uniquely ergodic.     

Almost automorphic symbolic minimal sets without unique ergodicity

Given a metrizable monothetic groupG with generatorg and a suitable closed nowhere dense subsetC of positive Haar measure, we associate a natural compact metric space whose points are almost automorphic symbolic minimal sets. It is then shown that those minimal sets which have positive topological entropy and fail to be uniquely ergodic form a esidual set. The example due to P. Julius [2] of a Toeplitz sequence of positive entropy which, is uniquely ergodic shows that the “residual” conclusion is sharp.     

Nonsingular dynamical systems,Bratteli diagrams and Markov odometers

We prove that any ergodic non-singular transformation is orbit equivalent to a Markov odometer which is uniquely ergodic.     

Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique

In [Rees, M., A minimal positive entropy homeomorphism of the 2-torus, J. London Math. Soc. 23 (1981) 537-550], Mary Rees has constructed a minimal homeomorphism of the n-torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotation and to control precisely the measurable dynamics of f. This yields in particular the following result: Any compact manifold of dimensiond?2which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy.More generally, given some homeomorphism R of a compact manifold and some homeomorphism hC of a Cantor set, we construct a homeomorphism f which “looks like” R from the topological viewpoint and “looks like” R×hC from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds?     

Minimal Cantor Systems and Unimodal Maps

We consider the restriction of unimodal maps f to the omega-limit set y ( c ) of the critical point for certain cases where y ( c ) is a Minimal Cantor set. We investigate the relation of these minimal systems to enumeration scales (generalized adding machines), to Vershik adic transformations on ordered Bratelli diagrams and to substitution shifts. Sufficient conditions are given for ( y ( c ), f ) to be uniquely ergodic.     

A dynamical systems approach to the Kadison-Singer problem

In these notes we develop a link between the Kadison-Singer problem and questions about certain dynamical systems. We conjecture that whether or not a given state has a unique extension is related to certain dynamical properties of the state. We prove that if any state corresponding to a minimal idempotent point extends uniquely to the von Neumann algebra of the group, then every state extends uniquely to the von Neumann algebra of the group. We prove that if any state arising in the Kadison-Singer problem has a unique extension, then the injective envelope of a C*-crossed product algebra associated with the state necessarily contains the full von Neumann algebra of the group. We prove that this latter property holds for states arising from rare ultrafilters and δ-stable ultrafilters, independent, of the group action and also for states corresponding to non-recurrent points in the corona of the group.     

Random walks on compact groups and the existence of cocycles

We show that certain skew products in ergodic theory are isomorphic to the shifts defined by random walks. We conclude the existence of cocycles for any finite measure preserving ergodic automorphism or flow, taking values in an arbitrary compact group, which determine ergodic skew products.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic. Received 5 March 2001; in revised form 16 August 2001     

The topology of group extensions of C systems

The paper is concerned with the topological and metric properties of group extensions of C systems. The basic theorem describes the topologically transitive component, the ergodic component, and the K component of a group extension of a C system. It is shown that each of these components is a group sub-bundle of a principal bundle in which the group extension acts. The frame flow on a manifold of negative curvature is seen to be a special case of a group of extension of a C system. It is shown that the space of frames on a compact three-dimensional manifold with negative curvature does not have any group sub-bundles, so that the frame flow on manifolds of this class is topologically transitive, ergodic, and a K system.     

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.     

The uniquely embeddable planar graphs

Whitney [7] proved in 1932 that for any two embeddings of a planar 3-connected graph, their combinatorial duals are isomorphic. In this manner, the term “uniquely embeddable planar graph” was introduced. It is a well-known fact that combinatorial and geometrical duals are equivalent concepts. In this paper, the concept of unique embeddability is introduced in terms of special types of isomorphisms between any two embeddings of a planar graph. From this, the class U of all graphs which are uniquely embeddable in the plane according to this definition, is determined, and the planar 3-connected graphs are a proper subset of U. It turns out that the graphs in U have a unique geometrical dual (i.e., for any two embeddings of such a graph, their geometrical duals are isomorphic). Furthermore, the theorems and their proofs do not involve any type of duals.     

Topological mixing and uniquely ergodic systems

Every ergodic transformation (X, T, ℬ,μ) has an isomorphic system (Y, U, ν) which is uniquely ergodic and topologically mixing. This work is a part of an M.Sc. thesis written at The Hebrew University of Jerusalem under the supervision of Professor B. Weiss to whom the author is greatly indebted.     

Skew products over rotations with exotic properties

We show that a $\text{ Z}_{2}$ skew product of a badly approximable rotation can be minimal and not uniquely ergodic. This construction is used to construct a Z skew product of a rotation where the orbit of a.e. point is dense but Lebesgue measure is not ergodic.     

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.     

Indecomposability of treed equivalence relations

We define rigorously a “treed” equivalence relation, which, intuitively, is an equivalence relation together with a measurably varying tree structure on each equivalence class. We show, in the nonamenable, ergodic, measure-preserving case, that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations.