On the Lévy constants of Sturmian continued fractions

Abstract

The L\'evy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic continued fraction expansion and it is well-known that this implies the existence of a L\'evy constant. Let $a, b$ be distinct positive integers. If the sequence of partial quotients of an irrational real number is a Sturmian sequence over $\{a, b\}$, then it has a L\'evy constant, which depends only on $a$, $b$, and the slope of the Sturmian sequence, but not on its intercept. We show that the set of L\'evy constants of irrational real numbers whose sequence of partial quotients is periodic or Sturmian is equal to the whole interval $[\log ((1+\sqrt 5)/2 ), + \infty)$.