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Ergodicity for piecewise smooth cocycles over toral rotations

Anzelm Iwanik (1998)

Fundamenta Mathematicae

Let α be an ergodic rotation of the d-torus 𝕋 d = ℝ d / ℤ d . For any piecewise smooth function f : 𝕋 d → ℝ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on L 2 ( 𝕋 d ) is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product S f : 𝕋 d + 1 → 𝕋 d + 1 must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum of V...

Estimates of capacity of self-similar measures

Jozef Myjak, Tomasz Szarek (2002)

Annales Polonici Mathematici

We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems ( S i , p i ) : i = 1 , . . . , N where S i are bi-lipschitzean transformations.

Existence of discrete ergodic singular transforms for admissible processes

Doğan Çömez (2008)

Colloquium Mathematicae

This article is concerned with the study of the discrete version of generalized ergodic Calderón-Zygmund singular operators. It is shown that such discrete ergodic singular operators for a class of superadditive processes, namely, bounded symmetric admissible processes relative to measure preserving transformations, are weak (1,1). From this maximal inequality, a.e. existence of the discrete ergodic singular transform is obtained for such superadditive processes. This generalizes the well-known...

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

We extend Champernowne’s construction of normal numbers to base b to the ℤ d case and obtain an explicit construction of a generic point of the ℤ d shift transformation of the set 0 , 1 , . . . , b - 1 ℤ d .

Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications

Tim Austin (2010)

Fundamenta Mathematicae

We study a generalized notion of a homogeneous skew-product extension of a probability-preserving system in which the homogeneous space fibres are allowed to vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of 'direct integral' for a 'measurable family' of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions,...

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