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Maximizing Bernoulli measures and dimension gaps for countable branched systems
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 7 / July 2021
- Published online by Cambridge University Press:
- 26 May 2020, pp. 1921-1939
- Print publication:
- July 2021
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- Article
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Kifer, Peres, and Weiss proved in [A dimension gap for continued fractions with independent digits. Israel J. Math.124 (2001), 61–76] that there exists
$c_{0}>0$, such that 
$\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$ for any probability measure 
$\unicode[STIX]{x1D707}$, which makes the digits of the continued fraction expansion independent and identically distributed 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.
