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arXiv · 1703.03164

A dimension gap for continued fractions with independent digits - the non stationary case

Ariel Rapaport
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Abstract

We show there exists a constant $0<c_{0}<1$ such that the dimension of every measure on $[0,1]$, which makes the digits in the continued fraction expansion independent, is at most $1-c_{0}$. This extends a result of Kifer, Peres and Weiss from 2001, which established this under the additional assumption of stationarity. For $k\ge1$ we prove an analogues statement for measures under which the digits form a $*$-mixing $k$-step Markov chain. This is also generalized to the case of $f$-expansions. In addition, we construct for each $k$ a measure, which makes the continued fraction digits a stationary and $*$-mixing $k$-step Markov chain, with dimension at least $1-2^{3-k}$.

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Ariel Rapaport. 2017-03-09. A dimension gap for continued fractions with independent digits - the non stationary case. https://arxiv.org/abs/1703.03164

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