Hitting functions for mixed partitions

Let $T_{\rho}$ be an irrational rotation on a unit circle $S^{1}\simeq [0,1)$. Consider the sequence $\{\mathcal{P}_{n}\}$ of increasing partitions $S^{1}$. Define hitting times $N_{n}(\mathcal{P}_n;x,y):= \inf\{j\geq 1\mid T^{j}_{\rho}(y)\in P_{n}(x)\}$, where $P_{n}(x)$ is element $\mathcal{P}_{n}$ containing $x$. D. Kim and B. Seo in [9] proved that rescaled $K_n(\mathcal{Q}_n;x,y):= \frac{\log N_n(\mathcal{Q}_n;x,y)}{n}$ a.e. (with respect to Lebesgue measure) converge $\log2$, $\{\mathcal{Q}_n\}$ associated with chaotic map $f_{2}(x):=2x \bmod 1$. The $f_{2}(x)$ has positive entropy $\log2$. A natural question what if $\{\mathcal{P}_n\}$ zero entropy. In present work we study behavior $K_n(\tau_n;x,y)$ mixed $\{\tau_{n}\}$ such $ \mathcal{P}_{n}\cap [0,\frac{1}{2}]$ $f_{2}$ $\mathcal{D}_{n}\cap [\frac{1}{2},1]$ $T_{\rho}$. It converges piecewise constant function two values. Also, it shown there are some rotations exhibit different behavior.