Some remarks on the metrical transitivity of invariant and quasi-invariant measures
Some Results on Affine Transformations of Compact Groups.
Some thoughts about Segal's ergodic theorem
Over fifty years ago, Irving Segal proved a theorem which leads to a characterization of those orthogonal transformations on a Hilbert space which induce ergodic transformations. Because Segal did not present his result in a way which made it readily accessible to specialists in ergodic theory, it was difficult for them to appreciate what he had done. The purpose of this note is to state and prove Segal's result in a way which, I hope, will win it the recognition which it deserves.
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.
Spectral properties of G-symbolic Morse shifts
Spectral properties of skew-product diffeomorphisms of tori
Spontaneous clustering in theoretical and some empirical stationary processes*
In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper...
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 of the densities of invariant measures for piecewise affine expanding non-renormalizable maps
Standardness of sequences of σ-fields given by certain endomorphisms
Let E be an ergodic endomorphism of the Lebesgue probability space X, ℱ, μ. It gives rise to a decreasing sequence of σ-fields A central example is the one-sided shift σ on with product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis defined on (X× Y, μ× ν) by . Here ℱ is the σ-field of μ× ν-measurable sets. Each field is a two-point extension of the one beneath it. Vershik has defined as “standard” any decreasing sequence of σ-fields isomorphic...
Stationary perturbations based on Bernoulli processes
Stochastic Stability in Some Chaotic Dynamical Systems.
Strict measure rigidity for unipotent subgroups of solvable groups.
Strict Uniformity in Ergodic Theory.
Stricte ergodicité des échanges d'intervalles.
Strictly ergodic, uniform positive entropy models
Strong estimate for square functions in higher dimensions.
Strongly Mixing g-Measures.
Strongly mixing sequences of measure preserving transformations
We call a sequence of measure preserving transformations strongly mixing if tends to for arbitrary measurable , . We investigate whether one can pass to a suitable subsequence such that almost surely for all (or “many”) integrable .
Structure of mixing and category of complete mixing for stochastic operators
Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak...