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For a real number $0<lambda<2$, we introduce a transformation $T_lambda$ naturally associated to expansion in $lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_lambda$ provides an algorithm to expand any positive real number in $lambda$-continued fraction. We prove the conjugacy between $T_lambda$ and some $beta$-shift, $beta>1$. Some properties of the map $lambdamapstobeta(lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,infty[$ but non-analytic.
Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.
We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. W
It is widely believed that the continued fraction expansion of every irrational algebraic number $alpha$ either is eventually periodic (and we know that this is the case if and only if $alpha$ is a quadratic irrational), or it contains arbitrarily la
This is a translation of Eulers Latin paper De fractionibus continuis observationes into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the Riccati-Differ