Dynamics of $λ$-continued fractions and $β$-shifts

Details Dynamics of $λ$-continued fractions and $β$-shifts Dynamics of $λ$-continued fractions and $β$-shifts

2011-03-31

arxiv.org/abs/1103.6181v1

Mathematics

Probability

Scientific paper

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 $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.

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