A NOTE ON THE RETURN TIME OF STURMIAN SEQUENCES
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Let Rn be the the ¯rst return time to its initial n- word. Then the Ornstein-Weiss ¯rst return time theorem implies that logRn divided by n converges to entropy. We consider the convergence of logRn for Sturmian sequences which has the lowest complexity. In this case, we normalize the logarithm of the ¯rst return time by log n. We show that for any numbers 1 · ®; ¯ · 1, there is a Sturmian sequence of which limsup is ® and liminf is 1=¯.
Abstract
Let Rn be the the ¯rst return time to its initial n- word. Then the Ornstein-Weiss ¯rst return time theorem implies that logRn divided by n converges to entropy. We consider the convergence of logRn for Sturmian sequences which has the lowest complexity. In this case, we normalize the logarithm of the ¯rst return time by log n. We show that for any numbers 1 · ®; ¯ · 1, there is a Sturmian sequence of which limsup is ® and liminf is 1=¯.
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