## Notes of Frederic Paulin’s Cambridge lecture 25-04-2017

On continued fractions, diagonal group actions and trees in positive characteristic

With Uri Shapira.

1. Continued fractions

Consider the ring of 1-variable polynomials ${R=F_q[Y]}$ over a finite field. Its fraction field ${K=F_q(Y)}$ can be completed with respect to the absolute value

$\displaystyle \begin{array}{rcl} |\frac{P}{Q}|=q^{deg(P)-deg(Q)} \end{array}$

into the field of formal Laurent series ${\hat K=F_q((Y^{-1}))}$. Elements of ${\hat K}$ are series which start with some negative index ${i_0}$. Elements for which ${i_0>0}$ form a subset ${M}$. They have an integral part, belonging to ${R}$, and a fractional part, which belongs to ${M}$. Hence a notion of continued fraction.

The Artin map maps ${f}$ to ${\{1/f\}}$ preserves a measure, which is ${1/q}$ times the Haar measure of ${\hat K/M}$.

Let ${QI}$ be the subset of quadratic irrationals in ${\hat K}$. Such elements have evetually periodic continued fraction expansion. Let ${r_f}$ denote the starting point of the periodic part, and ${n_f}$ the length of the period. We are interested in the distribution of finite orbits of Artin’s map. Hence we introduce the followingatomic measures

$\displaystyle \begin{array}{rcl} \nu_f=\frac{1}{n_f}\sum_{i=r_f}^{r_f+n_f-1}\Delta_{\psi^i(f)} \end{array}$

Our model is the following result in the classical setting (where ${R={\mathbb Z}}$).

Theorem 1 (Aka-Shapira) For every prime ${p\in R}$, for all ${f\in QI}$, ${\nu_{p^n f}}$ 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 ${p\in R}$ and ${f\in QI}$.

1. In the continued fraction expansion, one coefficient has degree much larger than others (in fact much larger than the sum of the others).
2. There exist uncountably many elements in ${PGL(2,R)}$ for which, for every ${f'\in QI}$, there exists ${c>0}$ such that among accumulation points of measures ${\nu_{p^n \gamma_n f}}$, there is a measure ${\theta>c\,\nu_{f'}}$.

In other words, equidistribution does not take place at all.

2. Proof

Study dynamics of diagonal matrices ${A}$ on ${X=\Gamma\setminus G}$, ${G=PGl(2,\hat K)}$, ${\Gamma=PGl(2,R)}$. One can view ${X}$ as the set of homothety equivalence classes of rank 2 ${R}$-submodules ${\hat K}$ generating ${\hat K^2}$.

Claim. If ${f\in QI}$, let

$\displaystyle \begin{array}{rcl} g_f=\begin{pmatrix} f^\sigma & f \\ 1 & 1 \end{pmatrix}, \end{array}$

where ${\sigma}$ is the automorphism of the quadratic field generated by ${f}$. The ${A}$-orbit of ${\Gamma g_f}$ is compact. Let ${\mu_f}$ denote the ${A}$-invariant probability measure on ${\Gamma g_f A}$, viewed as a measure on ${X}$.

Unlike in the classical case (where these measure converge to the Liouville measure),

Theorem 3 (Kemaishi-Paulin-Shapira) In any weak-${*}$ accumulation point of ${\mu_{p^n f}}$, there is a definite loss of mass.

2.1. Tools

Let ${\mathcal{O}}$ be the ring of formal power series (i.e. with ${i_0\geq 0}$). Let ${T}$ be the Bruhat-Tits tree of ${G=PGl(2,\hat K)}$. Its vertices correspond to equivalence classes of ${\mathcal{O}}$-lattices in ${\hat K^2}$. Links are projective lines over ${F_q}$. A ray can be labelled by a sequence of elements of ${F_q}$. The boundary is a projective line over ${\hat K}$, ${M}$ corresponds to the ends of the subtree whose first label is ${0}$.

The quotient of ${T}$ by ${\Gamma}$ is a half-line with stabilizers ${PGL_1(2,F_q)}$ for the origin and

$\displaystyle \begin{array}{rcl} \{\begin{pmatrix} a & b \\ 0 & \lambda \end{pmatrix}\,:\,a,\,\lambda\in F_q,\,deg(b)\leq n\} \end{array}$

for other vertices.

Serre observed that the projection of a geodesic ray in ${\Gamma\setminus T}$ keeps going back and forth.

Classically, the continued fraction expansion of an element ${f\in\hat K}$ accounts for depth of penetration of a geodesic ray converging to ${f}$ into horoballs based at points of ${K}$. In our setting, depth = degree of polynomial.

Our main technical tool is

Proposition 4 There exists a cross section of the action of ${A}$ on ${X}$, meeting every orbit in a countable set, and bijection with ${M}$ such that the first return map is the Artin map ${\psi}$. The push forward of ${\mu}$ is ${\nu}$.

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.