Smooth Non-Bernoulli K-Automorphisms.
Solvability of the functional equation f = (T-I)h for vector-valued functions
Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and in measure for some f ∈ M(μ;X), then also in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X) satisfying...
Some dynamical properties of S-unimodal maps
We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.
Some examples of cocycles with simple continuous singular spectrum
We study spectral properties of Anzai skew products defined by , 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 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 limit ratio theorem related to a real endomorphism in case of a neutral fixed point
Some new cases of realization of spectral multiplicity function for ergodic transformations
Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle
We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle . A set of integers is called -Bohr if it is recurrent for all products of rotations on , and Bohr if it is recurrent for all products of rotations on . It is a result due to Katznelson that for each there exist sets of integers which are -Bohr but not -Bohr. We present new examples of -Bohr sets which are not Bohr, thanks to a construction which...
Some notes on topological recurrence.
Some open problems in dynamical systems.
Some Remarks Concerning an Example of a Minimal, Non-Uniquely Ergodic Interval Exchange Transformation.
Some remarks on the metrical transitivity of invariant and quasi-invariant measures
Spectral and mixing properties of actions of amenable groups.
Spectral isomorphisms of Morse flows
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 , 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
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.
Spectres de M-extensions aléatoires
Spectrum of the Poincaré Map for Periodic Reflecting Rays in Generic Domains.
SRB measures for non-hyperbolic systems with multidimensional expansion
SRB-like Measures for C⁰ Dynamics
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
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?...
Stability of Orbit spaces of Endomorphisms.