On continued fractions, diagonal group actions and trees in positive characteristic
With Uri Shapira.
1. Continued fractions
Consider the ring of 1-variable polynomials over a finite field. Its fraction field can be completed with respect to the absolute value
into the field of formal Laurent series . Elements of are series which start with some negative index . Elements for which form a subset . They have an integral part, belonging to , and a fractional part, which belongs to . Hence a notion of continued fraction.
The Artin map maps to preserves a measure, which is times the Haar measure of .
Let be the subset of quadratic irrationals in . Such elements have evetually periodic continued fraction expansion. Let denote the starting point of the periodic part, and the length of the period. We are interested in the distribution of finite orbits of Artin’s map. Hence we introduce the followingatomic measures
Our model is the following result in the classical setting (where ).
Theorem 1 (Aka-Shapira) For every prime , for all , converges to the invariant smooth (Gauss) measure.
It turns out that this does not generalize at all to positive characteristic. Because of Frobenius automorphisms, raising primes to powers behaves unevenly.
Theorem 2 (Paulin-Shapira) Fix a prime and .
- In the continued fraction expansion, one coefficient has degree much larger than others (in fact much larger than the sum of the others).
- There exist uncountably many elements in for which, for every , there exists such that among accumulation points of measures , there is a measure .
In other words, equidistribution does not take place at all.
Study dynamics of diagonal matrices on , , . One can view as the set of homothety equivalence classes of rank 2 -submodules generating .
Claim. If , let
where is the automorphism of the quadratic field generated by . The -orbit of is compact. Let denote the -invariant probability measure on , viewed as a measure on .
Unlike in the classical case (where these measure converge to the Liouville measure),
Theorem 3 (Kemaishi-Paulin-Shapira) In any weak- accumulation point of , there is a definite loss of mass.
Let be the ring of formal power series (i.e. with ). Let be the Bruhat-Tits tree of . Its vertices correspond to equivalence classes of -lattices in . Links are projective lines over . A ray can be labelled by a sequence of elements of . The boundary is a projective line over , corresponds to the ends of the subtree whose first label is .
The quotient of by is a half-line with stabilizers for the origin and
for other vertices.
Serre observed that the projection of a geodesic ray in keeps going back and forth.
Classically, the continued fraction expansion of an element accounts for depth of penetration of a geodesic ray converging to into horoballs based at points of . In our setting, depth = degree of polynomial.
Our main technical tool is
Proposition 4 There exists a cross section of the action of on , meeting every orbit in a countable set, and bijection with such that the first return map is the Artin map . The push forward of is .
It turns out that, during excursions, unlike in the classical case, some amount of mass is lost. This is due to an arithmetic rather that geometric mechanism.