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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...

Standardness of sequences of σ-fields given by certain endomorphisms

Jacob Feldman, Daniel Rudolph (1998)

Fundamenta Mathematicae

 Let E be an ergodic endomorphism of the Lebesgue probability space X, ℱ, μ. It gives rise to a decreasing sequence of σ-fields ℱ , E - 1 ℱ , E - 2 ℱ , . . . A central example is the one-sided shift σ on X = 0 , 1 ℕ with 1 2 , 1 2 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 ( x , y ) → ( σ ( x ) , T x ( 1 ) ( y ) ) . 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...

Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends, Jörg Schmeling (2001)

Czechoslovak Mathematical Journal

We call a sequence ( T n ) of measure preserving transformations strongly mixing if P ( T n - 1 A ∩ B ) tends to P ( A ) P ( B ) for arbitrary measurable A , B . We investigate whether one can pass to a suitable subsequence ( T n k ) such that 1 K ∑ k = 1 K f ( T n k ) ⟶ ∫ f d P almost surely for all (or “many”) integrable f .

Structure of mixing and category of complete mixing for stochastic operators

Anzelm Iwanik, Ryszard Rębowski (1992)

Annales Polonici Mathematici

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...

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