Escape of mass of the Thue-Morse sequence

Abstract

Every Laurent series in $\mathbb{F}_q\left(\left(t^{-1}\right)\right)$ has a continued fraction expansion whose partial quotients are polynomials. De Mathan and Teulié proved that the degrees of the partial quotients of the left-shifts of every quadratic Laurent series are unbounded. Shapira and Paulin and Kemarsky improved this by showing that certain sequences of probability measures on the space of lattices in the plane $\mathbb{F}_q\left(\left(t^{-1}\right)\right)^2$ exhibit positive escape of mass and conjectured that this escape is full -- that is, that these probability measures converge to zero. We disprove this conjecture by analysing in detail the case of the Laurent series over $\mathbb{F}_2$ whose sequence of coefficients is the Thue-Morse sequence. The proof relies on the discovery of explicit symmetries in its number wall.