Théorème d'équidistribution de Brolin en dynamique p-adique

Abstract

Présenté par Jean-Christophe Yoccoz RésuméNous démontrons un analogue du théorème classique d'équidistribution de Brolin pour les applications rationnelles à une variable définies sur le corps p-adique C p . On construit une mesure invariante et mélangeante qui décrit la distribution (asymptotique) des préimages itérées d'un point donné. Cette mesure est à support dans l'espace analytique de P 1 (C p ), au sens de Berkovich, que l'on note P 1 (C p ). On démontre que le support de cette mesure est égale à l'ensemble de Julia dans P 1 (C p ), introduit par Rivera-Letelier. Nos résultats sont basés sur la notion d'opérateur de Laplace sur les arbres réels avec nombre arbitraire de branchements construit dans (C. Favre, M.

Heights and totally p-adic numbers

“…The proof of Theorem 1.1 is presented in Section 3. The main ingredient for a proof of the implication (i) ⇒ (ii) is an equidistribution theorem for points of small canonical height, independently achieved by Baker and Rumely [3], Favre and Rivera-Letelier [13], Chambert-Loir [9] and widely generalized by Yuan [28]. The converse implication follows mainly from deep results of Rivera-Letelier [24] on the structure of the Berkovich Julia set of a rational map.…”

The minimal resultant locus

“…Since ord(−2 + 3ip 1/2 ) = 0 when p is odd, it follows from Lemma 1.4 (or directly from formula (13)), that ordRes ϕ (·) is increasing in the direction of u 2 at ζ D(0,p 3/2 ) . A similar argument applies for u 3 .…”

“…, b d and det(γ) will also be rational over H(x, y). Comparing (12) and (13) we see that each affine piece of ordRes ϕ (·) has the form mt + c, where m is an integer in the range…”

“…If we write the numerator of ϕ η as a 3 z 3 + a 2 z 2 + a 1 z + a 0 , then ord(a 2 ) = 0, and it follows from (13) …”

Greatest common divisors of iterates of polynomials

“…Proofs of similar equalities over function fields appear in [GTZ11, BD11, GHT15, BD13, YZ16], Thus, we only give a sketch here. The idea is to apply equidistribution results such as those in [BR06,CL06,FRL04], all of which hold over both number fields and function fields of characteristic 0. For each place v of K, the λ i equidistribute with respect to the measures of maximal entropy µ f,v and µ g,v for f and g respectively at v. This implies that the local canonical heights h f,v and h g,v for f and g are equal to each other.…”