# American Institute of Mathematical Sciences

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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