October  2008, 20(4): 1111-1122. doi: 10.3934/dcds.2008.20.1111

On circle rotations and the shrinking target properties

1. 

Department of Mathematics, Brandeis University, Waltham, MA 02454, United States

Received  March 2007 Revised  August 2007 Published  January 2008

We generalize the monotone shrinking target property (MSTP) to the $s$-exponent monotone shrinking target property ($s$MSTP) and give a necessary and sufficient condition for a circle rotation to have $s$MSTP.
    Using another variant of MSTP, we obtain a new, very short, proof of a known result, which concerns the behavior of irrational rotations and implies a logarithm law similar to D. Sullivan's logarithm law for geodesics.
Citation: Jimmy Tseng. On circle rotations and the shrinking target properties. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1111-1122. doi: 10.3934/dcds.2008.20.1111
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