On the multiplicative ergodic theorem for uniquely ergodic systems

We consider the question of uniform convergence in the multiplicative ergodic theorem

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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.     

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.     

An application of number theory to ergodic theory and the construction of uniquely ergodic models

Using a combinatorial result of N. Hindman one can extend Jewett’s method for proving that a weakly mixing measure preserving transformation has a uniquely ergodic model to the general ergodic case. We sketch a proof of this reviewing the main steps in Jewett’s argument. To the memory of Shlomo Horowitz The research of this author was supported by the National Science Foundation (USA).     

Minimal rates of entropy convergence for completely ergodic systems

If (X,T) is a completely ergodic system, then there exists a positive monotone increasing sequence {a _n } _n ^1/∞ with lim _n →∞a _n =∞ and a positive concave functiong defined on [1, ∞) for whichg(x)/x ² isnot integrable such that for all nontrivial partitions α ofX into two sets.     

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.     

Strictly ergodic models for non-invertible transformations

We generalize a result of R. Jewett [J]: IfT is an ergodic measure preserving transformation on (X, Ω,λ),T not necessarily invertible, there exists a strictly ergodicS acting on (Y, Θ,ν), whereY is compact, such that (X, Ω,λ, T) is measure theoretically isomorphic to (Y, Θ,ν, S).     

Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett–Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.     

A divergent teichmüller geodesic with uniquely ergodic vertical foliation

We construct an example of a quadratic differential whose vertical foliation is uniquely ergodic and such that the Teichmüller geodesic determined by the quadratic differential diverges in the moduli space of Riemann surfaces. This research is partially supported by NSF grant DMS0244472.     

$$C_0$$-semigroups associated with uniquely ergodic Kantorovich modifications of operators

We extend to the context of \(L^p\) spaces and \(C_0\)-semigroups of operators our previous results from Heilmann and Ra?a (Positivity 21:897–910, 2017.  https://doi.org/10.1007/s11117-016-0441-1), concerning the eigenstructure and iterates of uniquely ergodic Kantorovich modifications of linking operators.     

Entropy formula of Pesin type for noninvertible random dynamical systems

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Compact quantum ergodic systems

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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?     

The Shannon-McMillan theorem for ergodic quantum lattice systems

We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on -lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.