Dynamics of rational maps on the projective line of the field of p - PowerPoint PPT Presentation

Dynamics of rational maps on the projective line of the field of p -adic numbers Lingmin LIAO (Universit´ e Paris-Est Cr´ eteil) (joint with Ai-Hua Fan , Shi-Lei Fan and Yue-Fei Wang ) International Conference on p -adic Mathematical Physics and its applications Mathematical Institute SANU Belgrade, September 8th 2015 Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 1/30

Outline Introduction 1 Polynomials on Z p 2 Rational maps of degree 1 on P 1 ( Q p ) 3 Rational maps with good reduction 4 Dynamics of φ ( x ) = ax + 1 /x, a ∈ Q p with p ≥ 3 5 Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 2/30

Introduction Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 3/30

I. p -adic dynamical systems A dynamical system is a couple ( X, T ) where T : X → X is a transformation on the space X . We call ( X, T ) a p -adic dynamical system if X is a p -adic space. The beginning : Oselies-Zieschang 1975 : automorphisms of Z p Herman-Yoccoz 1983 : complex p -adic dynamical systems Volovich 1987 : p -adic string theory Example : ( Z p , f ) with f ∈ Z p [ x ] being a polynomial. It is 1 -Lipschitz and then equicontinuous. The system ( X, T ) is equicontinuous if ∀ ǫ > 0 , ∃ δ > 0 s. t. d ( T n x, T n y ) < ǫ ( ∀ n ≥ 1 , ∀ d ( x, y ) < δ ) . Theorem Let X be a compact metric space and T : X → X be an equicontinuous transformation . Then the following statements are equivalent : (1) T is minimal (every orbit is dense). (2) T is uniquely ergodic (there is a unique invariant measure). (3) T is ergodic for any/some invariant measure with X as its support. Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 4/30

II. 1 -Lipschitz continuous dynamics on Z p For 1 -Lipschitz continuous maps f : Z p → Z p , the dynamical systems ( Z p , f ) are extensively studied. For example : Polynomials : Coelho-Parry 2001 : ax and distribution of Fibonacci numbers Gundlach-Khrennikov-Lindahl 2001 : ergodicity of x n on cycles. A. Fan-Li-Yao-Zhou 2007 : minimal decomposition of ax + b . Diarra-Sylla 2014 : periodic orbits of Chebyshev polynomials. S. Fan-Liao 2015 : minimal decomposition of x 2 . Mahler Series Anashin 1994, 1995, 1998, 2002. van der Put Series Yurova 2010 ; Anashin-Khrennikov-Yurova 2011, 2012, 2014 ; Khrennikov-Yurova 2011 ; Jeong 2012. T-functions Anashin-Khrennikov-Yurova 2014. Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 5/30

Polynomials on Z p Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 6/30

I. Polynomial dynamical systems on Z p Let f ∈ Z p [ x ] be a polynomial with coefficients in Z p . Polynomial dynamical systems : f : Z p → Z p , noted as ( Z p , f ) . Theorem (Ai-Hua Fan, L ; Adv. Math. 2011) minimal decomposition Let f ∈ Z p [ x ] with deg f ≥ 2 . The space Z p can be decomposed into three parts : Z p = P ⊔ M ⊔ B , where P is the finite set consisting of all periodic orbits ; M := ⊔ i ∈ I M i ( I finite or countable) → M i : finite union of balls, → f : M i → M i is minimal ; B is attracted into P ⊔ M . Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 7/30

II. Dynamics for each minimal part Given a positive integer sequence ( p s ) s ≥ 0 such that p s | p s +1 . Profinite groupe : Z ( p s ) := lim ← Z /p s Z . Odometer : The transformation τ : x �→ x + 1 on Z ( p s ) . Theorem (Chabert–A. Fan–Fares 2009) Let E be a compact set in Z p and f : E → E a 1 -lipschitzian transforma- tion. If the dynamical system ( E, f ) is minimal, then ( E, f ) is conjuguate to the odometer ( Z ( p s ) , τ ) where ( p s ) is determined by the structure of E . Theorem (Fan–L 2011 : Minimal components of polynomials) Let f ∈ Z p [ X ] be a polynomial and O ⊂ Z p a clopen set, f ( O ) ⊂ O . Suppose f : O → O is minimal. If p ≥ 3 , then ( O, f | O ) is conjugate to the odometer ( Z ( p s ) , τ ) where ( p s ) s ≥ 0 = ( k, kd, kdp, kdp 2 , . . . ) (1 ≤ k ≤ p, d | ( p − 1)) . If p = 2 , then ( O, f | O ) is conjugate to ( Z 2 , x + 1) . Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 8/30

III. Minimality on the whole space Z p Theorem (Larin 2002), General polynomials, only for p = 2 Let p = 2 and let f ( x ) = � a k x k ∈ Z 2 [ X ] be a polynomial. Then ( Z p , f ) is minimal iff a 0 ≡ 1 (mod 2) , a 1 ≡ 1 (mod 2) , 2 a 2 ≡ a 3 + a 5 + · · · (mod 4) , a 2 + a 1 − 1 ≡ a 4 + a 6 + · · · (mod 4) . General polynomials for p = 3 : Durand-Paccaut 2009 . Quadratic polynomials for all p : Larin 2002 + Knuth 1969 . Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 9/30

IV. Minimal decomposition of affine polynomials on Z p Let T a,b x = ax + b ( a, b ∈ Z p ) . Denote V = { z ∈ U : ∃ m ≥ 1 , s . t . z m = 1 } . U = { z ∈ Z p : | z | = 1 } , Easy cases : a ∈ Z p \ U ⇒ one attracting fixed point b/ (1 − a ) . 1 a = 1 , b = 0 ⇒ every point is fixed. 2 a ∈ V \ { 1 } ⇒ every point is on a ℓ -periodic orbit, with ℓ the 3 smallest integer � 1 such that a ℓ = 1 . Theorem (AH. Fan, MT. Li, JY. Yao, D. Zhou 2007) Case p ≥ 3 : a ∈ ( U \ V ) ∪ { 1 } , v p ( b ) < v p (1 − a ) ⇒ p v p ( b ) minimal parts. 4 a ∈ U \ V , v p ( b ) ≥ v p (1 − a ) ⇒ ( Z p , T a,b ) is conjugate to ( Z p , ax ) . 5 Decomposition : Z p = { 0 } ⊔ ⊔ n ≥ 1 p n U . (1) One fixed point { 0 } . (2) All ( p n U , ax )( n ≥ 0) are conjugate to ( U , ax ) . For ( U , T a, 0 ) : p v p ( a ℓ − 1) ( p − 1) /ℓ minimal parts, with ℓ the smallest integer � 1 such that a ℓ ≡ 1(mod p ) . Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 10/30

Two typical decompositions of Z p Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 11/30

V. One application in Number Theory Proposition (Fan-Li-Yao-Zhou 2007) Let k � 1 be an integer, and let a, b, c be three integers in Z coprime with � mod p k � p � 2 . Let s k be the least integer � 1 such that a s k ≡ 1 . (a) If b �≡ a j c (mod p k ) for all integers j ( 0 � j < s k ), then p k ∤ ( a n c − b ) , for any integer n � 0 . (b) If b ≡ a j c (mod p k ) for some integer j ( 0 � j < s k ), then we have N Card { 1 � n < N : p k | ( a n c − b ) } = 1 1 lim . s k N → + ∞ Remark : Consider T : x �→ ax . Then p k | ( a n c − b ) ⇔ | T n ( c ) − b | p ≤ p − k ⇔ T n ( c ) ∈ B ( b, p − k ) . Coelho and Parry 2001 : Ergodicity of p -adic multiplications and the distribution of Fibonacci numbers. Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 12/30

Rational maps of degree 1 on P 1 ( Q p ) Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 13/30

0. Rational maps on Q p x + a Mukhamedov and Rozikov 2004 : bx + c . ax 2 Khamraev and Mukhamedov 2006 : bx +1 . Dragovich, Khrennikov and Mihajlovi´ c 2007 : rational maps of degree 1 on the adelic space. Albeverio, Rozikov and Sattarov 2013 : (2 , 1) -rational maps on the field of p -adic complex numbers. Sattarov 2015 : (3 , 2) -rational maps on the field of p -adic complex numbers. Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 14/30

I. Projective line over Q p For ( x 1 , y 1 ) , ( x 2 , y 2 ) ∈ Q 2 p \ { (0 , 0) } , we say that ( x 1 , y 1 ) ∼ ( x 2 , y 2 ) if ∃ λ ∈ Q ∗ p s.t. x 1 = λx 2 and y 1 = λy 2 . Projective line over Q p : P 1 ( Q p ) := ( Q 2 p \ { (0 , 0) } ) / ∼ Spherical metric : for P = [ x 1 , y 1 ] , Q = [ x 2 , y 2 ] ∈ P 1 ( Q p ) , define | x 1 y 2 − x 2 y 1 | p ρ ( P, Q ) = max {| x 1 | p , | y 1 | p } max {| x 2 | p , | y 2 | p } . Viewing P 1 ( Q p ) as Q p ∪ {∞} , for z 1 , z 2 ∈ Q p ∪ {∞} we define | z 1 − z 2 | p ρ ( z 1 , z 2 ) = if z 1 , z 2 ∈ Q p , max {| z 1 | p , 1 } max {| z 2 | p , 1 } and � 1 , if | z | p ≤ 1 ; ρ ( z, ∞ ) = 1 / | z | p , if | z | p > 1 . Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 15/30

Geometric representations of P 1 ( Q 2 ) and P 1 ( Q 3 ) Lingmin LIAO University Paris-East Cr´ eteil Dynamics of rational maps on the projective line of Q p 16/30