Some applications of the Arithmetic euqidistribution theorem in Complex Dynamics
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1 Some applications of the Arithmetic euqidistribution theorem in Complex Dynamics! Hexi Ye University of British Columbia Seattle, Jan 9, 2016!!
2 Arithmetic euqidistribution Baker and Rumely, Favre and Rivera-Letelier established the arithmetic equidistribution of points with small P 1 height on.
3 Arithmetic euqidistribution Baker and Rumely, Favre and Rivera-Letelier established the arithmetic equidistribution of points with small P 1 height on. Chambert-Loir showed the equidistribution of points with small height on curves, while for general varieties the equidistribution theorem was given by Yuan.
4 Arithmetic euqidistribution For example: Let K be a number field and f : P 1 (K)! P 1 (K) be a rational function of degree greater than one.
5 Arithmetic euqidistribution For example: Let K be a number field and f : P 1 (K)! P 1 (K) be a rational function of degree greater than one. The canonical height for preperiodic points x of f. ĥ f (x) =0
6 Arithmetic euqidistribution For example: Let K be a number field and f : P 1 (K)! P 1 (K) be a rational function of degree greater than one. The canonical height ĥ f (x) =0 for preperiodic points x of f. Then Galois orbits of preperiodic points of f equidistribute on µ f entropy. P 1 w. r. t. the measure of maximal
7 Arithmetic euqidistribution For example: Let K be a number field and f : P 1 (K)! P 1 (K) be a rational function of degree greater than one. The canonical height ĥ f (x) =0 for preperiodic points x of f. Then Galois orbits of preperiodic points of f equidistribute on P 1 w. r. t. the measure of maximal µ f entropy. More generally, the preperiodic points of f : P n (K)! P n (K) equidistribute on w. r. t. the canonical measure. P n µ f
8 Arithmetic euqidistribution More examples: On the complex plane C, parameters c 2 C with f(z) =z 2 + c postcritcially finite (PCF, i.e. the critical point 0 is preperiodic), equidistribute w. r. t. the harmonic (bifurcation) measure of the Mandelbrot set.
9 Arithmetic euqidistribution More examples: On the complex plane C, parameters c 2 C with f(z) =z 2 + c postcritcially finite (PCF, i.e. the critical point 0 is preperiodic), equidistribute w. r. t. the harmonic (bifurcation) measure of the Mandelbrot set. Parameters c with z^2+c being PCF Picture by DeMarco
10 Arithmetic euqidistribution More examples: On the complex plane C, parameters c 2 C with f(z) =z 2 + c postcritcially finite (PCF, i.e. the critical point 0 is preperiodic), equidistribute w. r. t. the harmonic (bifurcation) measure of the Mandelbrot set. Parameters c with z^2+c being PCF Picture by DeMarco
11 Arithmetic euqidistribution and application More examples (Baker-DeMarco): Fixed an a 2 K, parameters c 2 C with a being preperiodic under f(z) =z d +c, equidistribute w.r.t. the bifurcation measure on C for the marked point a.
12 Arithmetic euqidistribution and application More examples (Baker-DeMarco): Fixed an a 2 K, parameters c 2 C with a being preperiodic under f(z) =z d +c, equidistribute w.r.t. the bifurcation measure on C for the marked point a. They use this to show that if a d 6= b d, then only finitely many c with both a and b preperiodic under f(z) =z d + c.
13 Arithmetic euqidistribution and application More examples (DeMarco-Wang-Y.): Fix, for family f t (z) = z z 2 + tz +1 the set of parameter t with critical points ±1 preperiodic, equidistributes on C w.r.t. the corresponding bifurcation measures. Parameters t with +1 preperiodic Parameters t with -1 preperiodic Pictures by DeMarco
14 Arithmetic euqidistribution and application More examples (DeMarco-Wang-Y.): Fix, for family f t (z) = z z 2 + tz +1 the set of parameter t with critical points ±1 preperiodic, equidistributes on C w.r.t. the corresponding bifurcation measures. We use this to show that when 6= 0, there are only finitely many PCF points in Per 1 ( ) M 2 (moduli space of quadratic maps).
15 Arithmetic euqidistribution and application More examples (DeMarco-Wang-Y.): Legendre family E t : {y 2 = x(x 1)(x t)}. Take a constant section P a (t) = (a, p a(a 1)(a t)) (a 2 K a constant). The parameters t 2 C, withp a (t) being torsion in E t,equidistributew.r.t. to some bifurcation measure. Pictures by DeMarco
16 Arithmetic euqidistribution and application More examples (DeMarco-Wang-Y.): Legendre family E t : {y 2 = x(x 1)(x t)}. Take a constant section P a (t) = (a, p a(a 1)(a t)) (a 2 K a constant). Use this, we get (Originally proved by Masser and Zannier) For a 6= b 2 K, there are only finitely many t with both P a (t) and P b (t) torsion in E t.
17 Arithmetic euqidistribution and application A question asked by Baker and DeMarco: In the moduli space of rational functions, which kind of variety contains Zariski dense PCF points?!
18 Arithmetic euqidistribution and application A question asked by Baker and DeMarco: In the moduli space of rational functions, which kind of variety contains Zariski dense PCF points?! An analogous result (Ghioca-Krieger-Nguyen-Y.): If non-vertical, non-horizontal curve C C 2 contains Zariski dense (a, b) with both z d + a and z d + b PCF, then C is given by y x =0with d 1 = 1. Ideas: (1) Equidistribution of small points (2) Combinatorial properties of the Mandelbrot set
19 Arithmetic euqidistribution and application Dynamical Manin-Mumford Conjecture (by Shouwu Zhang): The relation of preperiodic subvariety and subvariety with Zariski dense preperiodic points for a dynamic system.
20 Arithmetic euqidistribution and application Dynamical Manin-Mumford Conjecture (by Shouwu Zhang): The relation of preperiodic subvariety and subvariety with Zariski dense preperiodic points for a dynamic system. We (Ghioca-Nguyen-Y.) proved this conjecture for the case when the underline dynamic space is P 1 P 1 Ideas: (1) Equidistribution of small points (2) Symmetries of Julia set by Levin (3) Some other techniques
21 Thank you!
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