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Unique Bernoulli g -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a g -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique g -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the g -measure.

Uniqueness of a martingale-coboundary decomposition of stationary processes

Pavel Samek, Dalibor Volný (1992)

Commentationes Mathematicae Universitatis Carolinae

In the limit theory for strictly stationary processes f ∘ T i , i ∈ ℤ , the decomposition f = m + g - g ∘ T proved to be very useful; here T is a bimeasurable and measure preserving transformation an ( m ∘ T i ) is a martingale difference sequence. We shall study the uniqueness of the decomposition when the filtration of ( m ∘ T i ) is fixed. The case when the filtration varies is solved in [13]. The necessary and sufficient condition of the existence of the decomposition were given in [12] (for earlier and weaker versions of the results see [7])....

Van der Corput sets in ℤ d

Vitaly Bergelson, Emmanuel Lesigne (2008)

Colloquium Mathematicae

In this partly expository paper we study van der Corput sets in ℤ d , with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for d = 1 by Kamae and M. Mendès France and by Ruzsa, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some...

Waiting for long excursions and close visits to neutral fixed points of null-recurrent ergodic maps

Roland Zweimüller (2008)

Fundamenta Mathematicae

We determine, for certain ergodic infinite measure preserving transformations T, the asymptotic behaviour of the distribution of the waiting time for an excursion (from some fixed reference set of finite measure) of length larger than l as l → ∞, generalizing a renewal-theoretic result of Lamperti. This abstract distributional limit theorem applies to certain weakly expanding interval maps, where it clarifies the distributional behaviour of hitting times of shrinking neighbourhoods of neutral fixed...

Weak almost periodicity of L 1 contractions and coboundaries of non-singular transformations

Isaac Kornfeld, Michael Lin (2000)

Studia Mathematica

It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on L 1 is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex L 1 such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible...

Weakly mixing but not mixing quasi-Markovian processes

Zbigniew Kowalski (2000)

Studia Mathematica

Let (f,α) be the process given by an endomorphism f and by a finite partition α = A i i = 1 s of a Lebesgue space. Let E(f,α) be the class of densities of absolutely continuous invariant measures for skew products with the base (f,α). We say that (f,α) is quasi-Markovian if E ( f , α ) ⊂ g : ⋁ B i i = 1 s s u p p g = ⋃ i = 1 s A i × B i . We show that there exists a quasi-Markovian process which is weakly mixing but not mixing. As a by-product we deduce that the set of all coboundaries which are measurable with respect to the ’chequer-wise’ partition for σ × S, where σ is...

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