Exponential return times in a zero-entropy process (Q1947923)

The authors construct a finite-valued weakly mixing zero-entropy process with an exponential limit distribution of the hitting times to cylinder sets. More precisely, this process is of the form \((A^{\mathbb{N}},\mathcal{B},T,\mu)\), where \(A\) is a finite alphabet, \(\mathcal{B}\) is the standard \(\sigma\)-algebra generated by the cylinder sets \[ [x\langle n \rangle] = \left\{y\in A^{\mathbb{N}}: y_0=x_0,y_1=y_1,\dots,y_{n-1}=x_{n-1}\right\},\qquad x\in A^{\mathbb{N}},\;n\in\mathbb{N}, \] \(T\) is the shift map, and \(\mu\) a \(T\)-invariant probability measure. The hitting time to the cylinder set \([x\langle n\rangle]\) is the random variable on \(A^{\mathbb{N}}\) given by \[ \tau_{[x\langle n \rangle]}(y) = \min\left\{k\geq1 : T^k(y) \in [x\langle n \rangle] \right\}. \] The distribution function of the rescaled hitting time to \([x\langle n\rangle]\) is defined by \[ F_{x,n}(t) = \mu\left\{y\in A^{\mathbb{N}} : \mu([x\langle n\rangle])\tau_{[x\langle n\rangle]}(y) \leq t\right\}. \] The specific process constructed by the authors satisfies \[ \lim_{n\rightarrow\infty} F_{x,n}(t) = 1 - \mathrm{e}^{-t},\qquad t\geq0, \] for almost all \(x\). The same convergence result holds for the distribution functions of return times. In their proof, the authors compare the distributions of hitting and return times, and use a quite general result from \textit{N. Haydn, Y. Lacroix} and \textit{S. Vaienti} [Ann. Probab. 33, No. 5, 2043--2050 (2005; Zbl 1130.37305)].