Strong renewal theorems and Lyapunov spectra for \(\alpha \)-Farey and \(\alpha \)-Lüroth systems (Q2908168)

Some linearized generalizations of the classical Farey map and Gauss map are introduced and studied. Let \(\alpha=\{A_n : n\in \mathbb{N}\}\) be a countable partition of the unit interval, with \(A_n\) being non-empty left-open and right-closed intervals. Suppose that the atoms of \(\alpha\) are ordered from right to left and these atoms accumulate only at \(0\). Denote by \(a_n\) the length of \(A_n\), and write \(t_n=\sum_{k=n}^{\infty}a_k\). The \(\alpha\)-Farey map \(F_\alpha\) is defined by: \(F_\alpha(x)=(1-x)/a_1\) if \(x\in A_1\); \(F_\alpha(x)=a_{n-1}(x-t_{n+1})/a_n+t_n\) if \(x\in A_n\), \(n\geq 2\); and \(F_\alpha(0)=0\) if \(x=0\). The jump transformation \(L_\alpha\), called \(\alpha\)-Lüroth map, of \(F_\alpha\) to the interval \(A_1\) then can be explicitly given by: \(L_\alpha(0)=0\) and \(L_\alpha(x)=(t_n-x)/a_n\) if \(x\in A_n\), \(n\in\mathbb{N}\). Notice that if \(a_n=1/(n(n+1))\), the corresponding \(L_\alpha\) is nothing but the alternating Lüroth map.NEWLINENEWLINEFirst, the authors investigate dynamical properties of \(L_\alpha\) and \(F_\alpha\). It is proved that \(L_\alpha\) is measure-preserving and exact with respect to the Lebesgue measure. The map \(F_\alpha\) is also exact with respect to its unique (up to a constant) invariant measure which is absolutely continuous with respect to the Lebesgue measure. This invariant measure is \(\sigma\)-finite and is determined by the density function \(\sum_{n=1}^{\infty} (t_n/a_n)1_{A_n}\).NEWLINENEWLINEThen, using the renewal theorem of \textit{P. Erdös, W. Feller} and \textit{H. Pollard} [Bull. Am. Math. Soc. 55, 201--204 (1949; Zbl 0032.27802)], the authors establish weak and strong renewal laws for the sum-level sets for \(L_\alpha\), i.e., the sets of numbers with the sum of the first digits (of the \(\alpha\)-Lüroth expansion generated by \(L_\alpha\)) equal to a given level. These sum-level sets have been studied by the first and third authors of the present paper in the case of Gauss map [Discrete Contin. Dyn. Syst. 32, No. 7, 2437--2451 (2012; Zbl 1251.37013)].NEWLINENEWLINELast, applying some general thermodynamical results by \textit{J. Jaerisch} and the first author of this paper [Trans. Am. Math. Soc. 363, No. 1, 313--330 (2011; Zbl 1213.37038)], the authors obtain the Lyapunov spectra (the Hausdorff dimension spectra of the level sets of the Lyapunov exponents) of \(L_{\alpha}\) and \(F_\alpha\). The Lyapunov spectra of the classical Farey map and Gauss map were firstly obtained by the first and third authors of this paper [J. Reine Angew. Math. 605, 133--163 (2007; Zbl 1117.37003)]. These spectra are all obtained by using thermodynamical formalisms. However, another spectrum, sometimes named Besicovitch-Eggleston spectrum, concerning the frequencies of digits of the Lüroth expansion can be given by an explicit formula, see [\textit{A. Fan}, the reviewer, \textit{J. Ma} and \textit{B. Wang}, Nonlinearity 23, No. 5, 1185--1197 (2010; Zbl 1247.11104)]. The authors also give various examples and vivid graphs to illustrate different behaviours of the Lyapunov spectra of \(L_\alpha\) and \(F_\alpha\).