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  • Author or Editor: Z. Buczolich x
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Abstract  

One can define in a natural way irregular 1-sets on the graphs of several fractal functions, like Takagi’s function, Weierstrass-Cellerier type functions and the typical continuous function. These irregular 1-sets can be useful during the investigation of level-sets and occupation measures of these functions. For example, we see that for Takagi’s function and for certain Weierstrass-Cellerier functions the occupation measure is singular with respect to the Lebesgue measure and for almost every level the level set is finite.

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Abstract  

Suppose that f: ℝ → ℝ is a given measurable function, periodic by 1. For an α ∈ ℝ put M n α f(x) = 1/n+1 Σk=0 n f(x + ). Let Γf denote the set of those α’s in (0;1) for which M n α f(x) converges for almost every x ∈ ℝ. We call Γf the rotation set of f. We proved earlier that from |Γf| > 0 it follows that f is integrable on [0; 1], and hence, by Birkhoff’s Ergodic Theorem all α ∈ [0; 1] belongs to Γf. However, Γf\ℚ can be dense (even c-dense) for non-L 1 functions as well. In this paper we show that there are non-L 1 functions for which Γf is of Hausdorff dimension one.

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Abstract  

We construct a sequence (n k) such that n k + 1n k → ∞ and for any ergodic dynamical system (X, Σ, �, T) and f ε L 1(�) the averages

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\lim _{N \to \infty } (1/N)\sum\nolimits_{k = 1}^N {f(T^{n_k } x)}$$ \end{document}
converge to X f d� for � almost every x. Since the above sequence is of zero Banach density this disproves a conjecture of J. Rosenblatt and M. Wierdl about the nonexistence of such sequences.

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