Ontology type: schema:ScholarlyArticle
2016-12-19
AUTHORSShigeki Akiyama, Jonathan Caalim
ABSTRACTWe 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... »
PAGES357-370
http://scigraph.springernature.com/pub.10.1007/s00454-016-9849-4
DOIhttp://dx.doi.org/10.1007/s00454-016-9849-4
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