Statistics of return times for weighted maps of the interval (Q1577463)

Consider the following setting: \(T\) is a piecewise monotonic transformation (with \(b\) branches); \(T\) is piecewise \({\mathcal C}^2\), which means that there is a subdivision \((a_i)_{i=0}^{i=b}\) of \([0,1]\) such that \(T\) is monotonic and extends to a \({\mathcal C}^2\) map on each \((a_i,a_{i+1})\). Denote by \(\text{sing}(T)\) the set \(\{a_i\), \(i= 0,\ldots, b\}\) of the points where \(T\) is not continuous and let \(A_i = (a_i,a_{i+1})\). We call the set \(A_{i_1}^{i_n} = A_{i_1} \cap T^{-1}A_{i_2} \cap \cdots\cap T^{-n+1}A_{i_n}\) by \(n\)-cylinder. Denote by \({\mathcal P}^n\) the set of \(n\)-cylinders. For all \(x\) in \([0,1]\setminus \bigcup_{0}^\infty T^n(\text{sing}(T))\) and all \(n\), there is a unique \(n\)-cylinder containing \(x\), called \({\mathcal P}^n(x)\). For a measure \(\mu_\varphi\) invariant by \(T\), where \(\varphi\) is a measurable potential, a variable \(R_n\) is the first time when the associated process repeats its \(n\) first symbols. More strictly, \(R_n(x)=\inf\{k>0,\;T^k(x)\in{\mathcal P}^n(x)\}\). Define the entrance time in a cylinder \(A\) by \(\tau_A(x)=\inf\{k\geq 0,\;T^k(x)\in A\}\). The law of \(R_n\), correctly renormalized, converges to a lognormal distribution. This convergence strongly uses the fact that the law of the entrance time in a cylinder can be approximated by an exponential law.