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Some examples of cocycles with simple continuous singular spectrum

K. Frączek (2001)

Studia Mathematica

We study spectral properties of Anzai skew products T φ : ² → ² defined by T φ ( z , ω ) = ( e 2 π i α z , φ ( z ) ω ) , where α is irrational and φ: → is a measurable cocycle. Precisely, we deal with the case where φ is piecewise absolutely continuous such that the sum of all jumps of φ equals zero. It is shown that the simple continuous singular spectrum of T φ on the orthocomplement of the space of functions depending only on the first variable is a “typical” property in the above-mentioned class of cocycles, if α admits a sufficiently fast approximation....

Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle

Sophie Grivaux, Maria Roginskaya (2013)

Czechoslovak Mathematical Journal

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle 𝕋 . A set of integers is called r -Bohr if it is recurrent for all products of r rotations on 𝕋 , and Bohr if it is recurrent for all products of rotations on 𝕋 . It is a result due to Katznelson that for each r ≥ 1 there exist sets of integers which are r -Bohr but not ( r + 1 ) -Bohr. We present new examples of r -Bohr sets which are not Bohr, thanks to a construction which...

Spectral isomorphisms of Morse flows

T. Downarowicz, Jan Kwiatkowski, Y. Lacroix (2000)

Fundamenta Mathematicae

A combinatorial description of spectral isomorphisms between Morse flows is provided. We introduce the notion of a regular spectral isomorphism and we study some invariants of such isomorphisms. In the case of Morse cocycles taking values in G = ℤ p , where p is a prime, each spectral isomorphism is regular. The same holds true for arbitrary finite abelian groups under an additional combinatorial condition of asymmetry in the defining Morse sequence, and for Morse flows of rank one. Rank one is shown to...

Spectral properties of ergodic dynamical systems conjugate to their composition squares

Geoffrey R. Goodson (2007)

Colloquium Mathematicae

Let S and T be automorphisms of a standard Borel probability space. Some ergodic and spectral consequences of the equation ST = T²S are given for T ergodic and also when Tⁿ = I for some n>2. These ideas are used to construct examples of ergodic automorphisms S with oscillating maximal spectral multiplicity function. Other examples illustrating the theory are given, including Gaussian automorphisms having simple spectra and conjugate to their squares.

SRB-like Measures for C⁰ Dynamics

Eleonora Catsigeras, Heber Enrich (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

For any continuous map f: M → M on a compact manifold M, we define SRB-like (or observable) probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f always has observable measures, even if SRB measures do not exist. We prove that the definition of observability is optimal, provided that the purpose of the researcher is to describe the asymptotic statistics for Lebesgue almost all initial states. Precisely, the never empty set of all observable measures is...

Stability conditions of a queueing system model via fluid limits

Amina Angelika Bouchentouf (2013)

Applicationes Mathematicae

We study the ergodicity of a multi-class queueing model via fluid limits which have the advantage of describing the model in macroscopic form. We consider a model of processing bandwidth requests. Our system is defined by a network of capacity C=N, and a queue which contains an infinite number of items of various sizes 1, a' and b' with 1 < a' < b' < N. The problem considered is: Under what conditions on the parameters of some large classes of networks, do they reach the stationary regime?...

Stable ergodicity and julienne quasi-conformality

Charles Pugh, Michael Shub (2000)

Journal of the European Mathematical Society

In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably ergodic. Our main tools are the new concepts – julienne density point and julienne quasi-conformality of the stable and unstable holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points.

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