Polynomial and rational dynamical systems on the field of p-adic numbers and its projective line

Transcription

1 Polynomial and rational dynamical systems on the field of p-adic numbers and its projective line Lingmin LIAO (Université Paris-Est Créteil) (works with Ai-Hua Fan, Shi-Lei Fan, Yue-Fei Wang, Dan Zhou) Analysis on p-adic fields, p-adic Fuglede problem Institut de Mathématiques de Toulouse Toulouse, November 4th 2015 Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 1/35

2 Outline 1 Polynomials on Z p 2 Rational maps of degree 1 on P 1 (Q p ) 3 Rational maps with good reduction 4 p-adic repellers in Q p 5 Dynamics of φ(x) = ax + 1/x, a Q p with p 3 Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 2/35

3 Polynomials on Z p Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 3/35

4 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. Studies on p-adic dynamical systems : Oselies, Zieschang 1975 : automorphisms of the ring of p-adic integers Herman, Yoccoz 1983 : complex p-adic dynamical systems Volovich 1987 : p-adic string theory by applying p-adic numbers Thiran, Verstegen, Weyers 1989 Chaotic p-adic quadratic polynomials Lubin 1994 : iteration of analytic p-adic maps. Anashin 1994 : 1-Lipschitz transformation (Mahler series) Coelho, Parry 2001 : ax and distribution of Fibonacci numbers Gundlach, Khrennikov, Lindahl 2001 : x n Benedetto, Li Hua-Chieh, Rivera-Letelier, Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 4/35

5 II. 1-Lipschitz p-adic dynamical systems The system (X, T ) is equicontinuous if ɛ > 0, δ > 0 s. t. d(t n x, T n y) < ɛ ( n 1, d(x, y) < δ). Theorem : Let (X, T ) be equicontinuous. We have equivalences : (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. Theorem : If the continuous transformation T is uniquely ergodic (µ is the unique invariant probability measure), then for any continuous function g : X R, uniformly, Facts : n 1 1 g(t k (x)) n k=0 gdµ. 1-Lipschitz transformation is equicontinuous. Polynomial f Z p [x] : Z p Z p is equicontinuous. Rational maps having good reduction φ Q p (z) : P 1 (Q p ) P 1 (Q p ) is equicontinuous. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 5/35

6 III. One application in Number Theory Proposition (A. Fan Li Yao Zhou Adv. Math. 2007) Let k 1 be an integer, and let a, b, c be three integers in Z coprime with p 2. Let s k be the least integer 1 such that a s k 1 ( mod p k). (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 lim N + Remark : Consider T : x ax. Then 1 N Card{1 n < N : pk (a n c b)} = 1 s k. 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. Other motivations : cryptography, pseudo random numbers. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 6/35

7 IV. 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 Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 7/35

8 V. Dynamics for each minimal part Given a positive integer sequence (p s ) s 0 such that p s p s+1. Profinite groupe : Z (ps) := lim Z/p s Z. Odometer : The transformation τ : x x + 1 on Z (ps). Theorem (Chabert A. Fan Fares 2009) Let E be a compact set in Z p and f : E E a 1-lipschitzian transformation. If the dynamical system (E, f) is minimal, then (E, f) is conjuguate to the odometer (Z (ps), τ) where (p s ) is determined by the structure of E. Theorem (A. 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 (ps), 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 Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 8/35

9 VI. 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), 2a 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 Quadratic polynomials for all p : Larin Knuth Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 9/35

10 VII. Minimal decomposition of affine polynomials on Z p Let T a,b x = ax + b (a, b Z p ). Denote U = {z Z p : z = 1}, V = {z U : m 1, s.t. z m = 1}. Easy cases : 1 a Z p \ U one attracting fixed point b/(1 a). 2 a = 1, b = 0 every point is fixed. 3 a V \ {1} every point is on a l-periodic orbit, with l the smallest integer 1 such that a l = 1. Theorem (AH. Fan, MT. Li, JY. Yao, D. Zhou 2007) Case p 3 : 4 a (U \ V) {1}, v p (b) < v p (1 a) p vp(b) minimal parts. 5 a U \ V, v p (b) v p (1 a) (Z p, T a,b ) is conjugate to (Z p, ax). 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 vp(al 1) (p 1)/l minimal parts, with l the smallest integer 1 such that a l 1(mod p). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 10/35

11 Two typical decompositions of Z p Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 11/35

12 VIII. Some other works on minimal decomposition of polynomials on Z p A. Fan L (Adv. Math 2011) : minimal decomposition of quadratic polynomials for p = 2. S. Fan L (Proc. AMS 2016) : minimal decomposition of x 2 for all p. S. Fan L (J. Number Th. 2016) : minimal decomposition of Chebyshev polynomials for p = 2. S. Fan L (J. Diff. Equations 2015) : minimal decomposition of power series on the integral ring of a finite extensions of Q p. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 12/35

13 Rational maps of degree 1 on P 1 (Q p ) Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 13/35

14 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 ρ(p, Q) = x 1 y 2 x 2 y 1 p 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 ρ(z 1, z 2 ) = z 1 z 2 p max{ z 1 p, 1} max{ z 2 p, 1} if z 1, z 2 Q p, and ρ(z, ) = { 1, if z p 1 ; 1/ z p, if z p > 1. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 14/35

15 Geometric representations of P 1 (Q 2 ) and P 1 (Q 3 ) Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 15/35

16 II. Homographic maps Let φ(x) = ax + b with a, b, c, d Q p, ad bc 0, cx + d which induces an 1-to-1 map φ : P 1 (Q p ) P 1 (Q p ). The dynamics of φ depends on its fixed points which are the solution of ax + b cx + d = x cx2 + (d a)x b = 0. Discriminant : = (d a) 2 + 4bc. If = 0, then φ has only one fixed point x 0 in Q p and φ(x) is conjugate to a translation ψ(x) = x + α for some α Q p by g(x) = 1 x x 0. If 0 and Q p, then φ has two fixed points x 1, x 2 Q p and φ is conjugate to a multiplication x βx for some β Q p by g(x) = x x2 x x 1. If 0 and / Q p, then φ has no fixed point in Q p. But φ has two fixed points x 1, x 2 Q p ( ). So we will study the dynamics of φ on P 1 (Q p ( )) then its restriction on P 1 (Q p ). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 16/35

17 III. Minimal decomposition (φ admits no fixed point) Theorem (AH. Fan, SL. Fan, L, YF. Wang ; Adv. Math. 2014) Suppose that φ has no fixed points in P 1 (Q p ) and φ n id for each integer n > 0. Then 1 the system (P 1 (Q p ), φ) is decomposed as a finite number of minimal subsystems ; 2 these minimal subsystems are topologically conjugate to each other ; 3 the number of minimal subsystems is determined by the number λ := (a + d) + (a + d). Denote K = Q p ( ) be the quadratic extension of Q p generated by. π be an uniformizer of K : v p (π) = 1/e, where e is the ramification index of the extension. Define v π (x) := e v p (x) for x K. K be the residue field of K. l be the order in the group K of λ. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 17/35

18 IV. The case p 3 Theorem (FFLW, K = Q p ( N p ) is unramified (e = 1)) The dynamics (P 1 (Q p ), φ) is decomposed as ((p + 1)p vp(λl 1) 1 )/l minimal subsystems. Each subsystem is topologically conjugate to the adding machine on an odometer Z (ps) with (p s ) = (l, lp, lp 2, ). Theorem (FFLW, K = Q p ( p), Q p ( pn p ) is ramified (e > 1)) (1) If a + d p > p, then λ = 1 ( mod π). The dynamics (P 1 (Q p ), φ) is decomposed as 2p (vπ(λp 1) 3)/2 minimal subsystems. Moreover, each minimal subsystem is conjugate to the adding machine on the odometer Z (ps) with (p s ) = (1, p, p 2, ). (2) If a + d p < p, then λ = 1 ( mod π). The dynamics (P 1 (Q p ), φ) is decomposed as p (vπ(λp +1) 3)/2 minimal subsystems. Moreover, each minimal subsystem is conjugate to the adding machine on the odometer Z (ps) with (p s ) = (2, 2p, 2p 2, ). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 18/35

19 V. Minimal (ergodic) conditions Corollary (FFLW, case p 3) The system (P 1 (Q p ), φ) is minimal if and only if one of the following conditions satisfied (1) K = Q p ( ) is unramified, l = p + 1 and v p (λ l 1) = 1, (2) K = Q p ( ) is ramified and v π (λ p + 1) = 3. Corollary (FFLW, case p = 2) The system (P 1 (Q 2 ), φ) is minimal if and only if one of the following conditions satisfied (1) K = Q 2 ( ) = Q 2 ( 3), l = 3 and v 2 (λ 2l 1) = 2, (2) K = Q 2 ( ) = Q 2 ( 1), Q 2 ( 3), a + b 2 = 2 and v π (λ 2 + 1) = 2. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 19/35

20 Rational maps with good reduction Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 20/35

21 I. Rational maps with good reduction Let : Z p Z/pZ be the reduction modulo p defined by a ã a mod p. The reduction of a polynomial f(z) = n i=0 a iz i Z p [z] is f(z) = n ã i z i. i=0 A rational map φ(z) Q p (z) can be written as a quotient of polynomials f(z), g(z) Z p [z] having no common factors, such that at least one coefficient of f or g has absolute value 1. The reduction of φ is φ(z) = f(z) g(z) F p(z). If deg φ = deg φ, we say φ has good reduction, and if deg φ < deg φ, we say φ has bad reduction. Theorem (Silverman s book : The arithmetic of dynamical systems) If a rational map φ has good reduction then it is 1-Lipschitz continuous with respect to the spherical metric. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 21/35

22 II. Minimal criterion for good reductive rational maps If φ has good reduction, then φ induces a map from P 1 (F p ) to itself, where P 1 (F p ) is the projective line over F p Theorem (A. Fan S. Fan L.L Y. Wang, preprint) Let φ : P 1 (Q p ) P 1 (Q p ) be a rational map with good reduction and deg φ 2. Then the dynamical system (P 1 (Q p ), φ) is minimal if and only if the following conditions are satisfied (1) The reduction φ is transitive on P 1 (F p ). (2) (φ p+1 ) (0) = 1 (mod p). (3) For p = 2 or 3, φ p+1 (0) 0 p = 1/p and φ (p+1)p (0) 0 p = 1/p 2. For p 5, φ p+1 (0) 0 p = 1/p. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 22/35

23 III. Min. decomp. for rational maps with good reduction Theorem (Fan-Fan-L-Wang, preprint) Let φ Q p (z) be a rational map with good reduction and deg φ 2. We have the following decomposition where P 1 (Q p ) = P M B, 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, φ : M i M i is minimal ; B is attracted into P M. Further if E P 1 (Q p ) is a minimal clopen invariant set of φ, then φ : E E is conjugate to the adding machine on an odometer Z (ps), where (p s ) = (k, kd, kdp, kdp 2, ) with integers k and d such that 1 k p + 1 and d (p 1). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 23/35

24 IV. Minimal (ergodic) conditions for p = 2 Theorem (Fan Fan L Wang, preprint) Each map in Q 2 (z) is conjugate to a map of the form φ(z) = a 0 + a 1 z + + a n 1 z n 1 + z n b 1 z + + b n 1 z n 1 + z n with n 2 and a i, b j Q 2. Let a n = b n = 1 and set A φ := i 0 a i, B φ := j 1 b j, A φ,1 := i 0 a 2i+1, A φ,2 := i 0 a 4i+1 and A φ,3 := i 0 a 4i+3. If φ has good reduction and minimal, then a i, b j Z 2, for 0 i n 1 and 1 j n 1, a 0 1 mod 2, b 1 1 mod 2, A φ 2 mod 4, A φ,1 1 mod 2, B φ 1 mod 2, (H) a n 1 b n 1 1 mod 2, a 0 b 1 (a n 1 b n 1 )(A φ,2 A φ,3 )B φ +2(b 2 a 1 + a n 2 b n 2 + b n 1 + A φ,3 ) 1 mod 4. Conversely, (H) implies that φ is Lipschitz and minimal. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 24/35

25 V. Corollary and examples Corollary (Fan Fan L Wang, preprint) Let φ Q 2 (z) be a 1-Lipschitz rational map of degree 3. Then the dynamical system (P 1 (Q 2 ), φ) is not minimal. Corollary (Fan Fan L Wang, preprint) Let φ Q 2 (z) be a rational map of degree 4 having good reduction. Then the dynamical system (P 1 (Q 2 ), φ) is not minimal. Fact : A rational map of degree 2 is never minimal on P 1 (Q 2 ). Example 1 : Let p = 3 and φ(z) = 2z2 +2z+1 z 3 3z 2 +z+1. The dynamical system (P 1 (Q p ), φ) is minimal. Example 2 : Let p = 3 and φ(z) = 2z+3 (z 1)(z 2). The dynamical system (P 1 (Q p ), φ) is not minimal and we decompose P 1 (Q p ) as P 1 (Q p ) = B 1 (0) (P 1 (Q p ) \ B 1 (0)), where B 1 (0) is a minimal component of φ and the points in P 1 (Q p ) \ B 1 (0) are attracted to B 1 (0). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 25/35

26 VI. Ideas and methods Anashin 1994, Chabert A. Fan Fares 2009, Local-Global Lemma : Let X Z p be a compact set. f : X X is minimal f n : X/p n Z p X/p n Z p is minimal n 1. Desjardins and Zieve 1994, Ph.D thesis of Zieve 1996 : induction from level n to level n + 1. S. Fan and L 2015 : minimal decomposition of power series on finite extensions of Q p. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 26/35

27 p-adic repellers in Q p Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 27/35

28 I. Settings f : X Q p, X Q p compact open. Assume that 1 f 1 (X) X ; 2 X = i I B p τ (ci) (with some τ Z), i I, τi Z s.t. Define Julia set : f(x) f(y) p = p τ i x y p ( x, y B p τ (c i)). (1) J f = f n (X). n=0 We have f(j f ) J f. (X, J f, f) is called a p-adic weak repeller if all τ i 0 in (1), but at least one > 0. a p-adic repeller if all τ i > 0 in (1). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 28/35

29 For any i I, let I i := {j I : B j f(b i ) } = {j I : B j f(b i )}. Define A = (A i,j ) I I : A ij = 1 if j I i ; A ij = 0 otherwise. If A is irreducible, we say that (X, J f, f) is transitive. (Σ A, σ) be the corresponding subshift. For i, j I, i j, let κ(i, j) be the integer s.t. c i c j p = p κ(i,j). for x = (x 0, x 1,, x n, ), y = (y 0, y 1,, y n, ) Σ A, define d f (x, y) = p τx 0 τx 1 τx n 1 κ(xn,yn) (if n 0), d f (x, y) = p κ(x0,y0) (if n = 0) where n = n(x, y) = min{i 0 : x i y i }. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 29/35

30 II. Results Theorem (A. Fan L.L Y. Wang D. Zhou, 2007) Let (X, J f, f) be a transitive p-adic weak repeller with matrix A. Then the dynamics (J f, f, p ) is isometrically conjugate to the shift dynamics (Σ A, σ, d f ). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 30/35

31 III. Examples Example 1 Let c = c0 p τ Q p with c 0 p = 1 and τ 1. Define p-adic logistic map f c : Q p Q p X = p τ Z p (1 + p τ Z p ), J c = (X). n=0 f c n f c (x) = cx(x 1). Theorem (Fan L Wang Zhou, 2007) (J c, f c ) is conjugate to ({0, 1} N, σ). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 31/35

32 Example 2 Let a Z p, a 1 (mod p) and m 1 be an integer. Consider f m,a : Q p Q p : f m,a (x) = xp ax p m. I m,a = {0 k < p m : k p ak 0 (mod p m )} X m,a = k I m,a (k + p m Z p ), J = (X). Theorem (Fan L Wang Zhou, 2007) n=0 f m n (J, f m,a ) is conjugate to ({0,..., p 1} N, σ). Theorem (Woodcock and Smart 1998) (J, f 1,1 ) is conjugate to ({0,..., p 1} N, σ). Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 32/35

33 Dynamics of φ(x) = ax + 1/x, a Q p with p 3 Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 33/35

34 I. φ(x) = ax + 1/x, a Q p acting on P 1 (Q p ) = ˆQ p Note that φ(x) φ(y) = (a 1 xy )(x y) and φ (x) = a 1 x 2. We distinguish three case : (1) a p = 1 ; (2) a p > 1 ; (3) a p < 1. (1) a p = 1 : it is easy to see that φ has good reduction. The action of φ on ˆQ p is non-expanding. We investigate its minimal decomposition. (2) a p > 1 : If a / Q p, then J φ = and lim n φ n (x) =, x Q p. If a Q p, then J φ. φ has two repelling fixed points x 1 = 1 1 a and x 1 = 1 (J φ, φ) ( 2, σ) ; For x Q p \ J φ, lim n φ n (x) =. 1 a ; Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 34/35

35 II. φ(x) = ax + 1/x, a Q p acting on ˆQ p continued (3) a p < 1 : If v p (a) is odd, then J φ = {0, }. If v p (a) is even, we distinguish two cases : (a) a / Q p J φ = {0, } ; (b) if a Q p, 1 if a / Q p, J φ consists of two parts J 1 and J 2, with (J 1, φ) being topologically conjugate to a subshift of finite type and J 2 being the tail of J 1. 2 For the case a / Q p, there are still lots of work to do. Lingmin LIAO University Paris-East Créteil Polynomial and rational dynamical systems on Qp and P 1 (Qp) 35/35