Measures, annuli and dimensions

DOI10.1007/S00209-023-03230-9arXiv2111.09379MaRDI QIDQ6383338

Zoltán Buczolich, Stéphane Seuret

Publication date: 17 November 2021

Abstract: Given a Radon probability measure $mu$ supported in $mathbb{R}^d$, we are interested in those points $x$ around which the measure is concentrated infinitely many times on thin annuli centered at $x$. Depending on the lower and upper dimension of $mu$, the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of $mu$-measure $0$ or of $mu$-measure $1$. The measure concentration we study is related to bad points for the Poincar'e recurrence theorem and to the first return times to shrinking balls under iteration generated by a weakly Markov dynamical system. The study of thin annuli and spherical averages is also important in many dimension-related problems, including Kakeya-type problems and Falconer's distance set conjecture.

Mathematics Subject Classification ID

Dynamical aspects of measure-preserving transformations (37A05) Ergodicity, mixing, rates of mixing (37A25) Fractals (28A80) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) (37D25) Hausdorff and packing measures (28A78) Notions of recurrence and recurrent behavior in topological dynamical systems (37B20)

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