Dynamical systems on the field of p-adic numbers

Transcription

1 Dynamical systems on the field of p-adic numbers Lingmin LIAO (Université Paris-Est Créteil) Dynamical Systems Seminar Institute of Mathematics, Academia Sinica August 19th 2013 Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 1/45

2 Outline 1 The field Q p of p-adic numbers and p-adic dynamical systems 2 1-Lipschitz p-adic polynomial dynamical systems 3 p-adic repellers in Q p 4 p-adic homographic maps Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 2/45

3 The field Q p of p-adic numbers and p-adic dynamical systems Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 3/45

4 I. The p-adic numbers p 2 a prime number. n N, n = N i=0 a ip i (a i = 0, 1,, p 1) Ring Z p of p-adic integers : Z p x = i=0 a ip i. Field Q p of p-adic numbers : fraction field of Z p : Q p x = i=v(x) a ip i, ( v(x) Z). Absolute value : x p = p v(x), metric : d(x, y) = x y p. 3Z Z Z 3 Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 4/45

5 II. Arithmetic in Q p Addition and multiplication : similar to the decimal way. Carrying from left to right. Example : x = (p 1) + (p 1) p + (p 1) p 2 +, then x + 1 = 0. So, 1 = (p 1) + (p 1) p + (p 1) p x = (p 2) + (p 1) p + (p 1) p 2 +. We also have substraction and division. Then we can define polynomials and rational maps. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 5/45

6 III. A result of analysis on Q p f : X( Q p ) Q p is continuously differentiable at a X if the following exists : Lemma (Local rigidity lemma) lim (x,y) (a,a),x y f(x) f(y). x y Let U be a clopen set and a U. Suppose f : U Q p is continuously differentiable, f (a) 0. Then there exists r > 0 such that B r (a) U and ( x, y B r (a)). f(x) f(y) p = f (a) p x y p Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 6/45

7 IV. Equicontinuous dynamics T : X X is equicontinuous if Theorem ɛ > 0, δ > 0 s. t. d(t n x, T n y) < ɛ ( n 1, d(x, y) < δ). 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. (2) T is uniquely ergodic. (3) T is ergodic for any/some invariant measure with X as its support. Fact : 1-Lipschitz transformation is equicontinuous. Fact : Polynomial f Z p [x] : Z p Z p is equicontinuous. 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, n 1 1 g(t k (x)) n k=0 gdµ. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 7/45

8 V. One motivation Distribution of recurrence sequence. Proposition (Fan-Li-Yao-Zhou 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 + 1 N Card{1 n < N : pk (a n c b)} = 1 s k. Coelho and Parry 2001 : Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 8/45

9 VI. Study on p-adic dynamical dystems 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 Dynamical systems on the field of p-adic numbers 9/45

10 1-Lipschitz p-adic polynomial dynamical systems Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 10/45

11 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 2011) minimal decomposition Let f Z p [x] with deg f 2. The space Z p can be decomposed into three parts : Z p = A B C, where A is the finite set consisting of all periodic orbits ; B := i I B i (I finite or countable) B i : finite union of balls, f : B i B i is minimal ; C is attracted into A B. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 11/45

12 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 (ps) := lim Z/p s Z. Odometer : The transformation τ : x x + 1 on Z (ps). Theorem (Chabert-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 (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 Dynamical systems on the field of p-adic numbers 12/45

13 III. Problems Problem 1 : Under what condition one has A = C =, and B admits a unique component? (i.e., f : Z p Z p is minimal?) Problem 2 : If (Z p, f) is not minimal, how to find the complete decomposition? Problem 3 : The solution for deg f = 2? (What about the affine polynomial case?) Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 13/45

14 IV. Minimality on the space Z p Theorem (Larin Knuth 1969), Quadratic polynomials, all p Let f(x) = ax 2 + bx + c with a, b, c Z p. f is minimal iff 1 a 0 (mod p), b 1 (mod p), c 0 (mod p), if p 5, 2 a 0 (mod 9), b 1 (mod 3), c 0 (mod 3) or ac 6 (mod 9), b 1 (mod 3), c 0 (mod 3), if p = 3. 3 a 0 (mod 2), a + b 1 (mod 4) and c 0 (mod 2), if p = 2. 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). Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 14/45

15 Minimality of general polynomials-criterion for the case p = 3 Let f(x) = a k x k Z 3 [X]. We can suppose a 0 = 1 : Note A 0 = a 0 0 (mod 3) f is not minimal. a 0 0 (mod 3) f is conjugate to a polynomial g with a 0 = 1. i 2N,i 0 a i, A 1 = i 1+2N a i, D 0 = i 2N,i 0 ia i, D 1 = i 1+2N Theorem (Durand-Paccaut 2009), General polynomials, only for p = 3 The system (Z 3, f) is minimal iff f satisfies A 0 3Z 3, A Z 3 and one of the following conditions : (1) D 0 0, D 1 2, a 1 1 [3] and A , 3a j>0 a 5+6j [9] ; (2) D 0 0, D 1 1, a 1 1 [3] and A , 6a j>0 a 2+6j [9] ; (3) D 0 1, D 1 0, a 1 2 [3] and A , 6a j>0 a 5+6j [9] ; (4) D 0 2, D 1 0, a 1 2 [3] and A , 3a j>0 a 2+6j [9]. ia i. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 15/45

16 V. 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 Dynamical systems on the field of p-adic numbers 16/45

17 Two typical decompositions of Z p Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 17/45

18 Theorem (Fan-Li-Yao-Zhou 2007) Case p = 2 : 4 a (U \ V) {1}, v p (b) < v p (1 a). v p (b) = 0 p vp(a+1) 1 minimal parts. v p (b) > 0 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 ) : 2 v2(a2 1) 2 minimal parts. Remark : For the case p = 2, all minimal parts (except for the periodic orbits) are conjugate to (Z 2, x + 1). Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 18/45

19 VI. Affine polynomials on Q p Let ϕ be an affine map defined by ϕ(x) = ax + b (a, b Q p, a 0, (a, b) (1, 0)). If a 1 : easy! For a = 1, we have the following conjugacy : a 1 : ax + b Q p Qp x b 1 a Q p ax x Q p b 1 a a = 1 : x + b Q p Qp x b Q p x + 1 x b Q p Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 19/45

20 VII. Affine polynomials on Q p -continued Theorem (AH. Fan, Y. Fares 2011) If K = Q p, then 1 ϕ(x) = x + 1 : Q p = Z p p n U. n=1 Z p is minimal. p n U contains p n 1 (p 1) minimal balls with radius 1. 2 ϕ(x) = ax (a is not a root of unity) : Q p = {0} 0 is fixed. All subsystems on p n U are conjugate to (U, ϕ). n Z p n U. For (U, ϕ) : (1) Case p 3 : p vp(al 1) (p 1)/l minimal balls of same radius, with l the smallest integer 1 such that a l 1(mod p). (2) Case p = 2 : 2 v 2(a 2 1) 2 minimal balls of same radius. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 20/45

21 Two typical decompositions of Q p Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 21/45

22 VIII. Minimal decomp. for quadratic polynomials on Z 2 Let f(x) := Ax 2 + Bx + C (A, B, C Z 2, A 0). Note := (B 1) 2 4AC. Then f admits fixed points iff Z 2. We distinguish three cases : B 0 (mod 2) B 1 (mod 2) and f admits fixed points ( Z 2 ) B 1 (mod 2) and f does not admit any fixed point ( Z 2 ) Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 22/45

23 Case 1 : B 0 (mod 2) There exist α, β, λ Z 2 such that αx + β Theorem (Fan-L 2011, x 2 λ) For x 2 λ, λ Z 2, we have Ax 2 + Bx + C Z 2 Z2 Z 2 x 2 λ Z 2 αx + β 1 λ 0 (mod 4) two attracting fixed points, basins : 2Z 2, 1 + 2Z 2 ; 2 λ 1 (mod 4) one attracting periodic orbit of period 2 ; 3 λ 2 (mod 4) two attracting fixed points, basins : 2Z 2, 1 + 2Z 2 ; 4 λ 3 (mod 4) one attracting periodic orbit of period 2. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 23/45

24 Case 2 : B 1 (mod 2) and f admits fixed points f admits fixed points, then α, β, b Z 2, such that αx + β Ax 2 + Bx + C Z 2 Z2 Z 2 x 2 + bx Z 2 αx + β Theorem (Fan-L 2011) We obtain the complete decomposition of x 2 + bx in Z 2. (There are countable infinite minimal component.) Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 24/45

25 Look at a special case. Theorem (Fan-L 2011, case b = 1) For (Z 2, x 2 + x), 1 The ball 1 + 2Z 2 is mapped into 2Z 2. 2 The ball 2Z 2 can be decomposed as : 2Z 2 = {0} 2 n n Z 2, n 2 and for each n 2, 2 n n Z 2 is decomposed as 2 n 2 minimal component : 2 n 1 + t2 n + 2 2n 2 Z 2, t = 0,..., 2 n 2 1. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 25/45

26 decomposition of x 2 + x 1 0 (mod 2) 0 2 (mod 2 2 ) 0 4 (mod 2 3 ) (mod 2 4 ) (mod 2 5 ) (mod 2 6 ) Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 26/45

27 Case 3 : B 1 (mod 2) and f has no fixed point Then α, β, d Z 2, but, d Z2, such that Theorem (Fan-L 2011) αx + β Ax 2 + Bx + C Z 2 Z2 Z 2 x 2 + x d Z 2 αx + β We obtain the complete decomposition of x 2 + x d ( d Z 2 ) in Z 2. Look at a special case. Theorem (Fan-L, 2011, case d 3 (mod 4) ) Let f(x) = x 2 + x d with d Z 2 and d 3 (mod 4). Then f 1+2Z2, is minimal and 2Z 2 is mapped into 1 + 2Z 2 by f. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 27/45

28 p-adic repellers in Q p Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 28/45

29 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 Dynamical systems on the field of p-adic numbers 29/45

30 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 Dynamical systems on the field of p-adic numbers 30/45

31 II. Results Theorem (Fan-L-Wang-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 Dynamical systems on the field of p-adic numbers 31/45

32 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 Theorem (Fan-L-Wang-Zhou, 2007) f c (x) = cx(x 1). (J c, f c ) is conjugate to ({0, 1} N, σ). Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 32/45

33 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 Dynamical systems on the field of p-adic numbers 33/45

34 p-adic homographic maps Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 34/45

35 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 : Let 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 K { }, 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 Dynamical systems on the field of p-adic numbers 35/45

36 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 ). φ( d/c) =, and φ( ) = a/c. φ is a composition of φ 1 (x) = αx, φ 2 (x) = x + β, φ 3 (x) = x 1/x. if D(a, r) is a disk in P 1 (Q p ), then φ(d(a, r)) is also a disk. (a disk in P 1 (Q p ) is a disk in Q p or a complement of a disk in Q p.) 1 φ 1 (D(a, r)) = D(αa, r α p ). 2 φ 2 (D(a, r)) = D(a + β, r). 3 if 0 D(a, r), φ 3 (D(a, r)) = P 1 (K) \ D(0, 1/r). 4 if 0 / D(a, r), φ 3 (D(a, r)) = D(a 1, r a 2 p ). Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 36/45

37 III. Fixed points and dynamics 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 Dynamical systems on the field of p-adic numbers 37/45

38 IV. Notations K is a finite extension of Q p. Still denote by p the extended absolute value of K. Degree : d = [K : Q p ]. Ramification index : e Valuation function : v p (x) := log p ( x p ). Im(v p ) = 1 e Z. O K := {x K : x p 1} : the local ring of K, P K := {x K : x p < 1} : its maximal ideal. Residual field : K = O K /P K. Then K = F p f, with f = d/e. Quadratic extensions : 7 quadratic extensions of Q 2 : Q 2 ( 1), Q 2 ( ±2), Q 2 ( ±3), Q 2 ( ±6). 3 quadratic extensions of Q p (p 3) : Q p ( p), Q p ( N p ), Q p ( pn p ), where N p is the smallest quadratic non-residue module p. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 38/45

39 V. Uniformizer and representation An element π K is a uniformizer if v p (π) = 1/e. Define v π (x) := e v p (x) for x K. Then Im(v π ) = Z, and v π (π) = 1. Let C = {c 0, c 1,..., c pf 1} be a fixed complete set of representatives of the cosets of P K in O K. Then every x K has a unique π-adic expansion of the form x = i=i 0 a i π i, where i 0 Z and a i C for all i i 0. Example : For Q p ( p) (p 3), take π = p, and x = a 0 + a 1 p + a2 p + a 3 p 3/2 + a 4 p 2 +. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 39/45

40 VI. Minimal decomposition (φ admits no fixed point) Theorem (AH. Fan, SL. Fan, L, YF. Wang (preprint)) 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 Denote λ := (a + d) + (a + d). K = Q p ( ) be the quadratic extension of Q p generated by. π be an uniformizer of K K be the residue field of K. l be the order in the group K of λ. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 40/45

41 VII. The case p 3 Theorem (Fan-Fan-L-Wang, K = Q p ( N p ) is unramified) 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 (Fan-Fan-L-Wang, K = Q p ( p), Q p ( pn p ) is ramified) (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 Dynamical systems on the field of p-adic numbers 41/45

42 VIII. The case p = 2 Theorem (FFLW, K = Q 2 ( 3) is unramified) The dynamical system (P 1 (Q 2 ), φ) is decomposed as 3 2 v2(λ2l 1) 2 /l minimal subsystems. Moreover, each minimal system is conjugate to the adding machine on the odometer Z (ps) with (p s ) = (l, l2, l2 2, ). Theorem (FFLW, K = Q 2 ( 2), Q 2 ( 2), Q 2 ( 6), Q 2 ( 6) ramified) (1) If a + d 2 > 2, then v π (λ 1) 3 is odd and the dynamical system (P 1 (Q 2 ), φ) is decomposed as 2 (vπ(λ 1) 1)/2 minimal subsystems. (2) If a + d 2 < 2, then v π (λ + 1) 3 is odd and the dynamical system (P 1 (Q 2 ), φ) is decomposed as 2 (vπ(λ+1) 1)/2 minimal subsystems. Moreover, each minimal system is conjugate to the adding machine on the odometer Z (ps) with (p s ) = (1, 2, 2 2, ). Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 42/45

43 IX. The case p = 2 (continued) Theorem (FFLW, K = Q 2 ( 1), Q 2 ( 3) is ramified) (1) If a + d 2 = 2, v π (λ 2 + 1) 2 is even and the system (P 1 (Q 2 ), φ) is decomposed as 2 (vπ(λ2 +1) 2)/2 minimal subsystems. (2) If a + d 2 > 2, then v π (λ 1) 4 is even and the system (P 1 (Q 2 ), φ) is decomposed as 2 vπ(λ 1)/2 minimal subsystems. (3) If a + d 2 < 2, v π (λ + 1) 4 is even and the system (P 1 (Q 2 ), φ) is decomposed as 2 vπ(λ+1)/2 minimal subsystems. Moreover, each minimal system is conjugate to the adding machine on the odometer Z (ps) with (p s ) = (1, 2, 2 2, ). Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 43/45

44 X. 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 Dynamical systems on the field of p-adic numbers 44/45

45 XI. Conjugacy and restriction Let x 1, x 2 be the two fixed points in K \ Q p. Let g(x) = x x2 x x 1. Denote ˆK = P 1 (K). Remark that ˆQ p = P 1 (Q p ) is invariant under φ. ( ˆQ p ) ˆK φ ( ˆQp ) ˆK g ˆK λx g ˆK Step I : Do minimal decomposition of (K, λx). Step II : Find g( ˆQ p ) and determine the restriction (g( ˆQ p ), λx). Step III : Go back to ˆQ p. Lingmin LIAO University Paris-East Créteil Dynamical systems on the field of p-adic numbers 45/45