Maximising Bernoulli measures and dimension gaps for countable branched systems

TL;DR

This paper identifies the measure with maximal dimension among those with i.i.d. continued fraction digits and extends the results to general countable branched systems, revealing dimension gaps.

Contribution

It establishes the existence of a measure with maximal dimension within a class of i.i.d. digit measures and generalizes to countable branched systems.

Findings

Existence of a measure with maximal dimension in the i.i.d. continued fraction class.

Dimension is bounded away from 1 by a positive constant.

Results extend to general countable branched systems.

Abstract

Kifer, Peres, and Weiss proved that there exists $c_0>0,$ such that $\dim \mu\leq 1-c_0$ for any probability measure $\mu$ which makes the digits of the continued fraction expansion i.i.d. random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.