A shrinking target theorem for ergodic transformations of the unit interval
نویسندگان
چکیده
<p style='text-indent:20px;'>We show that for <i>any</i> ergodic Lebesgue measure preserving transformation <inline-formula><tex-math id="M1">\begin{document}$ f: [0,1) \rightarrow $\end{document}</tex-math></inline-formula> and decreasing sequence id="M2">\begin{document}$ \{b_i\}_{i=1}^{\infty} of positive real numbers with divergent sum, the set</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {\underset{n=1}{\overset{\infty}{\cap}} \, {\underset{i=n}{\overset{\infty}{\cup}}}\,} f^{-i}(B (R_{\alpha}^{i} x,b_i)) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>has full almost every id="M3">\begin{document}$ x \in id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>. Here id="M5">\begin{document}$ B(x,r) is ball radius id="M6">\begin{document}$ r centered at id="M7">\begin{document}$ id="M8">\begin{document}$ R_{\alpha}: rotation by id="M9">\begin{document}$ As a corollary, we provide partial answer to question asked Chaika (Question id="M10">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>, [<xref ref-type="bibr" rid="b2">2</xref>]) in context interval exchange transformations.</p>
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ژورنال
عنوان ژورنال: Discrete and Continuous Dynamical Systems
سال: 2022
ISSN: ['1553-5231', '1078-0947']
DOI: https://doi.org/10.3934/dcds.2022042