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

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 $1-2^{3-k}$.