The Hausdorff dimension of recurrent sets in symbolic spaces
Let \((\Sigma,\sigma)\) be the one-sided shift space on \(N\) symbols \(1,2,\dots, N\) \((N\geq 2)\) with the usual metric \(d(x,y)= N^{-\inf\{k\geq 0: x_{k+1}\neq y_{k+1}\}}\) for \(x= (x_i)^\infty_{i=1}\) and \(y= (y_i)^\infty_{i=1}\). For any \(x= (x_i)^\infty_{i=1}\in \Sigma\) and positive integer \(n\) define NEWLINE\[NEWLINER_n(x)= \inf\{j\geq n: x_1\times x_2\cdots\times x_n= x_{j+1} x_{j+2}\times\cdots\times x_{j+n}\}.NEWLINE\]NEWLINE The authors show that for any \(\alpha,\beta\in [0,\infty]\) with \(\alpha\leq\beta\) the Hausdorff dimension of the set \(E_{\alpha,\beta}\) defined by NEWLINE\[NEWLINEE_{\alpha,\beta}= \Biggl\{x\in \Sigma: \liminf_{n\to\infty} {\log R_n(x)\over n}= \alpha,\;\limsup_{n\to\infty} {\log R_n(x)\over n}= \beta\Biggr\}NEWLINE\]NEWLINE is equal to one. To this end the authors construct Cantor-like subsets of \(E_{\alpha,\beta}\) so that their Hausdorff dimensions converge to one.
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