Invariant Measure of Rotational Beta Expansion and Tarski’s Plank Problem View Full Text


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Article Info

DATE

2016-12-19

AUTHORS

Shigeki Akiyama, Jonathan Caalim

ABSTRACT

We study the invariant measures of a piecewise expanding map in Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^m$$\end{document} defined by an expanding similitude modulo a lattice. Using the result of Bang (Proc Am Math Soc 2:990–993, 1951) on the plank problem of Tarski, we show that when the similarity ratio is at least m+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m+1$$\end{document}, the map has an absolutely continuous invariant measure equivalent to the m-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case m=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=2$$\end{document}, we obtain an alternative proof of the result in Akiyama and Caalim (J Math Soc Japan 69:1–19, 2016) together with some improvement. More... »

PAGES

357-370

References to SciGraph publications

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  • 2013-12-11. On the invariant density of the random β-transformation in ACTA MATHEMATICA HUNGARICA
  • 2000-01. Absolutely Continuous Invariant Measures¶for Piecewise Real-Analytic Expanding Maps¶on the Plane in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1985. Generalized bounded variation and applications to piecewise monotonic transformations in PROBABILITY THEORY AND RELATED FIELDS
  • 1957-09. Representations for real numbers and their ergodic properties in ACTA MATHEMATICA HUNGARICA
  • 1960-09. On theβ-expansions of real numbers in ACTA MATHEMATICA HUNGARICA
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  • 1964-03. Representations for real numbers in ACTA MATHEMATICA HUNGARICA
  • 2000-12. Absolutely continuous invariant measures for multidimensional expanding maps in ISRAEL JOURNAL OF MATHEMATICS
  • 2009-06-18. On the Characterization of Expansion Maps for Self-Affine Tilings in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2001-02. Absolutely continuous invariant measures for expanding piecewise linear maps in INVENTIONES MATHEMATICAE
  • 1999-04. Self-Replicating Tiles and Their Boundary in DISCRETE & COMPUTATIONAL GEOMETRY
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    URI

    http://scigraph.springernature.com/pub.10.1007/s00454-016-9849-4

    DOI

    http://dx.doi.org/10.1007/s00454-016-9849-4

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