The recurrence time for irrational rotations
Dong Han Kim · Osaka City University (Osaka City University) · 2006
Let $T$ be a measure preserving transformation on $X \\subset \\mathbb{R}^d$ with a Borel measure $\\mu$ and $R_E$ be the first return time to a subset $E$. If $(X,\\mu)$ has positive pointwise dimension for almost every $x$, then for almost every $x$ \\[ \\limsup_{r \\to 0^+} \\frac{\\log R_{B(x,r)}(x)}{-\\log \\mu(B(x,r))} \\le 1, \\] where $B(x,r)$ the the ball centered at $x$ with radius $r$. But the above property does not hold for the neighborhood of the `skewed' ball. Let $B(x,r;s) = (x - r^s, x + r)$ be an interval for $s >0$. For arbitrary $\\alpha \\ge 1$ and $\\beta \\ge 1$, there are uncountably many irrational numbers whose rotation satisfy that \\[ \\limsup_{r \\to 0^+} \\frac{\\log R_{B(x,r;s)}(x)}{-\\log \\mu (B(x,r;s))} = \\alpha \\quad \\text{and}\\quad \\liminf_{r \\to 0^+} \\frac{\\log R_{B(x,r;s)}(x)}{-\\log \\mu (B(x,r;s))} = \\frac{1}{\\beta} \\] for some $s$.