Dynamics of \(\lambda\)-continued fractions and \(\beta\)-shifts (Q1948757)

Let \(0<\lambda <2\) be a real number. The authors introduce and study a transformation \(T_{\lambda}\) associated to \(\lambda\)-continued fractions with alternating signs of partial quotients. The work is motivated by the previous works of the same authors [Probab. Theory Relat. Fields 142, No. 3--4, 619--648 (2008; Zbl 1146.37035); Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 1, 135--158 (2010; Zbl 1201.37091)], where the special cases \(\lambda_k= 2 \cos(\pi/k)\), i.e., the Rosen continued fractions, were studied and were applied to find the growth rate of random Fibonacci sequences. Define homographic functions \(h\) and \(h_0\) by \[ h(y):={1 \over \lambda -y} \quad \text{and} \quad h_0(y):={y \over \lambda y +1}. \] Let \(m_0^{\lambda}:=0\) and recursively \(m_{i+1}^{\lambda}:=h(m_i^{\lambda})\) if \(m_i^{\lambda}< \lambda\). One sees that \((m_i^{\lambda})\) is increasing and there exists \(i_\lambda \geq 1\) such that \(m_{i_\lambda}^\lambda\geq \lambda\). Define now recursively \((h_i)_{0\leq i \leq i_\lambda}\) by \(h_{i+1}:=h\circ h_i\). Then the following intervals form a partition of \([0,\infty[\): \[ I_i^\lambda:=[m_i^{\lambda}, m_{i+1}^{\lambda}[ \quad \text{for} \;0\leq i < i_\lambda, \quad \text{and} \quad I_{i_\lambda}^\lambda:=[m_{i_\lambda}^\lambda, \infty[. \] The transformation \(T_\lambda\) on \([0, \infty[\) is defined as \[ \forall x \in I_i^\lambda, \;T_\lambda(x):=h_i^{-1}(x). \] The definition of \(T_\lambda\) induces a symbolic coding of the orbits under \(T_\lambda\) on the alphabet \(\{0,1, \dots, i_\lambda\}\): \[ \forall x \in [0,\infty[, \quad \omega_{\lambda}(x):= x_0x_1x_2\dots, \] where \(x_{\ell}\) is the unique element in \(\{0,1, \dots, i_\lambda\}\) such that \(T^{\ell}_\lambda(x)\in I_{x_\ell}^\lambda\). The authors of the present paper prove that if a real number \(x\in [0,\infty[\) has the infinite coding \(x_0x_1x_2\dots\), which can be written as \(0^{e_0}a_00^{e_1}a_1\dots 0^{e_i}a_i \dots\), with \(a_i>0\) and \(e_i\geq 0\), then \[ x= [[ \underbrace{e_0+1,1,\dots, 1}_{a_0 \;\text{terms}},\underbrace{e_1+2,1,\dots, 1}_{a_1 \;\text{terms}}, \dots, \underbrace{e_i+2,1,\dots, 1}_{a_i \;\text{terms}},\dots ]]{}_{\lambda}, \] with the notation \[ [[ b_1,b_2, \dots, b_i, \dots]]{}_\lambda:=\cfrac{ 1 }{ b_1\lambda+\cfrac{ 1 }{ -b_2\lambda+\ddots \cfrac{ 1 }{ (-1)^{i-1}b_i\lambda+\dots.}}} \] Defining the coding of the infinity by \(\omega_\lambda(\infty)=\lim_{x\to\infty}\omega_\lambda(x)\), the authors show that a sequence \((x_n)_{n\geq 0}\) codes the orbit of a real number \(x\) if and only if for all \(n\geq 0\), the sequence \(x_nx_{n+1}\dots\) is strictly less than \(\omega_\lambda(\infty)\) in the lexicographic order. This is a property similar to the coding in the classical \(\beta\)-transformation. By this property, the authors obtain a conjugacy between \(T_\lambda\) and some \(\beta\)-transformation, and then establish the correspondence \(\lambda \mapsto \beta(\lambda)\), which is proved to be an increasing and continuous but not analytic map from \(]0,2[\) onto \(]1,\infty[\). As corollary, the entropy of \(T_\lambda\) is \(\log \beta(\lambda)\). The authors also give a geometrical interpretation of \(T_\lambda\) involving the rotation of angles. At the end of paper, the authors ask some open questions such as the description of the absolutely continuous \(T_\lambda\)-invariant measure and the relation between the algebraic properties of \(\lambda\) and the dynamical properties of \(T_\lambda\).