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

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Outline

1

Introduction

2

Polynomials on Zp

3

Rational maps of degree 1 on P1(Qp)

4

Rational maps with good reduction

5

Dynamics of φ(x) = ax + 1/x, a ∈ Qp with p ≥ 3

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Introduction

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• 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 Zp Herman-Yoccoz 1983 : complex p-adic dynamical systems Volovich 1987 : p-adic string theory Example : (Zp, f) with f ∈ Zp[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 nx, T ny) < 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.

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• II. 1-Lipschitz continuous dynamics on Zp

For 1-Lipschitz continuous maps f : Zp → Zp, the dynamical systems (Zp, f) are extensively studied. For example : Polynomials : Coelho-Parry 2001 : ax and distribution of Fibonacci numbers Gundlach-Khrennikov-Lindahl 2001 : ergodicity of xn 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 x2.

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.

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Polynomials on Zp

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• I. Polynomial dynamical systems on Zp

Let f ∈ Zp[x] be a polynomial with coefficients in Zp. Polynomial dynamical systems : f : Zp → Zp, noted as (Zp, f). Theorem (Ai-Hua Fan, L ; Adv. Math. 2011) minimal decomposition Let f ∈ Zp[x] with deg f ≥ 2. The space Zp can be decomposed into three parts : Zp = P ⊔ M ⊔ B, where P is the finite set consisting of all periodic orbits ; M := ⊔i∈IMi (I finite or countable) → Mi : finite union of balls, → f : Mi → Mi is minimal ; B is attracted into P ⊔ M.

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• II. Dynamics for each minimal part

Given a positive integer sequence (ps)s≥0 such that ps|ps+1. Profinite groupe : Z(ps) := lim

← Z/psZ.

Odometer : The transformation τ : x → x + 1 on Z(ps). Theorem (Chabert–A. Fan–Fares 2009) Let E be a compact set in Zp 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(ps), τ) where (ps) is determined by the structure of E. Theorem (Fan–L 2011 : Minimal components of polynomials) Let f ∈ Zp[X] be a polynomial and O ⊂ Zp 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 (ps)s≥0 = (k, kd, kdp, kdp2, . . . ) (1 ≤ k ≤ p, d|(p − 1)). If p = 2, then (O, f|O) is conjugate to (Z2, x + 1).

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• III. Minimality on the whole space Zp

Theorem (Larin 2002), General polynomials, only for p = 2 Let p = 2 and let f(x) = akxk ∈ Z2[X] be a polynomial. Then (Zp, f) is minimal iff a0 ≡ 1 (mod 2), a1 ≡ 1 (mod 2), 2a2 ≡ a3 + a5 + · · · (mod 4), a2 + a1 − 1 ≡ a4 + a6 + · · · (mod 4). General polynomials for p = 3 : Durand-Paccaut 2009. Quadratic polynomials for all p : Larin 2002 + Knuth 1969.

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• IV. Minimal decomposition of affine polynomials on Zp

Let Ta,bx = ax + b (a, b ∈ Zp). Denote U = {z ∈ Zp : |z| = 1}, V = {z ∈ U : ∃m ≥ 1, s.t. zm = 1}. Easy cases :

1

a ∈ Zp \ 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 aℓ = 1. Theorem (AH. Fan, MT. Li, JY. Yao, D. Zhou 2007) Case p ≥ 3 :

4

a ∈ (U \ V) ∪ {1}, vp(b) < vp(1 − a) ⇒ pvp(b) minimal parts.

5

a ∈ U \ V, vp(b) ≥ vp(1 − a) ⇒ (Zp, Ta,b) is conjugate to (Zp, ax). Decomposition : Zp = {0} ⊔ ⊔n≥1pnU. (1) One fixed point {0}. (2) All (pnU, ax)(n ≥ 0) are conjugate to (U, ax). For (U, Ta,0) : pvp(aℓ−1)(p − 1)/ℓ minimal parts, with ℓ the smallest integer 1 such that aℓ ≡ 1(mod p).

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Two typical decompositions of Zp

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• 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 p 2. Let sk be the least integer 1 such that ask ≡ 1

• mod pk

. (a) If b ≡ ajc (mod pk) for all integers j (0 j < sk), then pk ∤ (anc − b), for any integer n 0. (b) If b ≡ ajc (mod pk) for some integer j (0 j < sk), then we have lim

N→+∞

1 N Card{1 n < N : pk | (anc − b)} = 1 sk . Remark : Consider T : x → ax. Then pk | (anc − 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.

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Rational maps of degree 1 on P1(Qp)

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• 0. Rational maps on Qp

Mukhamedov and Rozikov 2004 :

x+a bx+c.

Khamraev and Mukhamedov 2006 :

ax2 bx+1.

Dragovich, Khrennikov and Mihajlovi´ c 2007 : rational maps of degree 1

• n the adelic space.

Albeverio, Rozikov and Sattarov 2013 : (2, 1)-rational maps on the field

• f p-adic complex numbers.

Sattarov 2015 : (3, 2)-rational maps on the field of p-adic complex numbers.

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• I. Projective line over Qp

For (x1, y1), (x2, y2) ∈ Q2

p \ {(0, 0)}, we say that (x1, y1) ∼ (x2, y2) if

∃λ ∈ Q∗

p s.t.

x1 = λx2 and y1 = λy2. Projective line over Qp : P1(Qp) := (Q2

p \ {(0, 0)})/ ∼

Spherical metric : for P = [x1, y1], Q = [x2, y2] ∈ P1(Qp), define ρ(P, Q) = |x1y2 − x2y1|p max{|x1|p, |y1|p} max{|x2|p, |y2|p}. Viewing P1(Qp) as Qp ∪ {∞}, for z1, z2 ∈ Qp ∪ {∞} we define ρ(z1, z2) = |z1 − z2|p max{|z1|p, 1} max{|z2|p, 1} if z1, z2 ∈ Qp, and ρ(z, ∞) = 1, if |z|p ≤ 1 ; 1/|z|p, if |z|p > 1.

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Geometric representations of P1(Q2) and P1(Q3)

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• II. Homographic maps

Let φ(x) = ax + b cx + d with a, b, c, d ∈ Qp, ad − bc = 0, which induces an 1-to-1 map φ : P1(Qp) → P1(Qp). 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 x0 in Qp and φ(x) is conjugate to a translation ψ(x) = x + α for some α ∈ Qp by g(x) =

1 x−x0 .

If ∆ = 0 and √ ∆ ∈ Qp, then φ has two fixed points x1, x2 ∈ Qp and φ is conjugate to a multiplication x → βx for some β ∈ Qp by g(x) = x−x2

x−x1 .

If ∆ = 0 and √ ∆ / ∈ Qp, then φ has no fixed point in Qp. But φ has two fixed points x1, x2 ∈ Qp( √ ∆). So we will study the dynamics of φ on P1(Qp( √ ∆)) then its restriction on P1(Qp).

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• 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 P1(Qp) and φn = id for each integer n > 0. Then

1

the system (P1(Qp), 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 = Qp( √ ∆) be the quadratic extension of Qp generated by √ ∆. π be an uniformizer of K : vp(π) = 1/e, where e is the ramification index of the extension. Define vπ(x) := e · vp(x) for x ∈ K. K be the residue field of K. ℓ be the order in the group K∗ of λ.

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• IV. The case p ≥ 3

Theorem (FFLW, K = Qp(

• Np) is unramified (e = 1))

The dynamics (P1(Qp), φ) is decomposed as ((p + 1)pvp(λℓ−1)−1)/ℓ minimal subsystems. Each subsystem is topologically conjugate to the adding machine on an odometer Z(ps) with (ps) = (ℓ, ℓp, ℓp2, · · · ). Theorem (FFLW, K = Qp(√p), Qp(

• pNp) is ramified (e > 1))

(1) If |a + d|p > | √ ∆|p, then λ = 1 ( mod π). The dynamics (P1(Qp), φ) is decomposed as 2p(vπ(λp−1)−3)/2 minimal subsystems. Moreover, each minimal subsystem is conjugate to the adding machine

• n the odometer Z(ps) with (ps) = (1, p, p2, · · · ).

(2) If |a + d|p < | √ ∆|p, then λ = −1 ( mod π). The dynamics (P1(Qp), φ) is decomposed as p(vπ(λp+1)−3)/2 minimal subsystems. Moreover, each minimal subsystem is conjugate to the adding machine

• n the odometer Z(ps) with (ps) = (2, 2p, 2p2, · · · ).

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• V. Minimal (ergodic) conditions

Corollary (FFLW, case p ≥ 3) The system (P1(Qp), φ) is minimal if and only if one of the following conditions satisfied (1) K = Qp( √ ∆) is unramified, ℓ = p + 1 and vp(λℓ − 1) = 1, (2) K = Qp( √ ∆) is ramified and vπ(λp + 1) = 3. Corollary (FFLW, case p = 2) The system (P1(Q2), φ) is minimal if and only if one of the following conditions satisfied (1) K = Q2( √ ∆) = Q2(√−3), ℓ = 3 and v2(λ2ℓ − 1) = 2, (2) K = Q2( √ ∆) = Q2(√−1), Q2( √ 3), |a + b|2 = | √ ∆|2 and vπ(λ2 + 1) = 2.

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Rational maps with good reduction

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• I. Rational maps with good reduction

Let · : Zp → Z/pZ be the reduction modulo p defined by a → a ≡ a mod p. The reduction of a polynomial f(z) = n

i=0 aizi ∈ Zp[z] is

• f(z) =

n

• i=0
• aizi.

A rational map φ(z) ∈ Qp(z) can be written as a quotient of polynomials f(z), g(z) ∈ Zp[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) ∈ Fp(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) A rational map φ has good reduction if and only if it is 1-Lipschitz conti- nuous with respect to the spherical metric.

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• II. Minimal criterion for good reductive rational maps

If φ has good reduction, then φ induces a map from P1(Fp) to itself, where P1(Fp) is the projective line over Fp Theorem (Fan-Fan-L-Wang, preprint) Let φ : P1(Qp) → P1(Qp) be a rational map with good reduction and deg φ ≥ 2. Then the dynamical system (P1(Qp), φ) is minimal if and only if the following conditions are satisfied (1) The reduction φ is transitive on P1(Fp). (2) (φp+1)′(0) = 1 (mod p). (3) For p = 2 or 3, |φp+1(0) − 0|p = 1/p and |φ(p+1)p(0) − 0|p = 1/p2. For p ≥ 5, |φp+1(0) − 0|p = 1/p.

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• III. Min. decomp. for rational maps with good reduction

Theorem (Fan-Fan-L-Wang, preprint) Let φ ∈ Qp(z) be a rational map with good reduction and deg φ ≥ 2. We have the following decomposition P1(Qp) = P ⊔ M ⊔ B, where P is the finite set consisting of all periodic orbits ; M := ⊔i∈IMi (I finite or countable) → Mi : finite union of balls, → φ : Mi → Mi is minimal ; B is attracted into P ⊔ M. Further if E ⊂ P1(Qp) is a minimal clopen invariant set of φ, then φ : E → E is conjugate to the adding machine on an odometer Z(ps), where (ps) = (k, kd, kdp, kdp2, · · · ) with integers k and d such that 1 ≤ k ≤ p + 1 and d | (p − 1).

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• IV. Minimal (ergodic) conditions for p = 2

Theorem (Fan-Fan-L-Wang, preprint) Each map in Q2(z) is conjugate to a map of the form φ(z) = a0 + a1z + · · · + an−1zn−1 + zn b1z + · · · + bn−1zn−1 + zn with n ≥ 2 and ai, bj ∈ Q2. Let an = bn = 1 and set Aφ :=

i≥0 ai,

Bφ :=

j≥1 bj, Aφ,1 := i≥0 a2i+1, Aφ,2 := i≥0 a4i+1 and

Aφ,3 :=

i≥0 a4i+3. Then φ has good reduction and (P1(Q2), φ) is

minimal if and only if                    ai, bj ∈ Z2, for 0 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 1, a0 ≡ 1 mod 2, b1 ≡ 1 mod 2, Aφ ≡ 2 mod 4, Aφ,1 ≡ 1 mod 2, Bφ ≡ 1 mod 2, an−1 − bn−1 ≡ 1 mod 2, a0b1(an−1 − bn−1)(Aφ,2 − Aφ,3)Bφ +2(b2 − a1 + an−2 − bn−2 + bn−1 + Aφ,3) ≡ 1 mod 4.

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• V. Corollaries

Corollary (Fan-Fan-L-Wang, preprint) Let φ ∈ Q2(z) be a rational map of degree 2 or 3 having good reduction. Then the dynamical system (P1(Q2), φ) is not minimal. Corollary (Fan-Fan-L-Wang, preprint) We can find all rational maps of order 4 with good reduction which are minimal on P1(Q2). Example 1 : Let p = 3 and φ(z) = −

2z2+2z+1 z3−3z2+z+1. The dynamical system

(P1(Qp), φ) is minimal. Example 2 : Let p = 3 and φ(z) =

2z+3 (z−1)(z−2). The dynamical system

(P1(Qp), φ) is not minimal and we decompose P1(Qp) as P1(Qp) = B1(0)

• (P1(Qp) \ B1(0)),

where B1(0) is a minimal component of φ and the points in P1(Qp) \ B1(0) are attracted to B1(0).

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• VI. Ideas and methods

Anashin 1994, Chabert–A. Fan–Fares 2009, Local-Global Lemma : Let X ⊂ Zp be a compact set. f : X → X is minimal ⇔ fn : X/pnZp → X/pnZp 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 Qp.

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Dynamics of φ(x) = ax + 1/x, a ∈ Qp with p ≥ 3

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• I. φ(x) = ax + 1/x, a ∈ Qp acting on P1(Qp) = ˆ

Qp

Note that φ(x) − φ(y) = (a − 1 xy )(x − y) and φ′(x) = a − 1 x2 . 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

• n ˆ

Qp is non-expanding. We investigate its minimal decomposition. (2) |a|p > 1 : If √−a / ∈ Qp, then Jφ = ∅ and limn→∞ φn(x) = ∞, ∀x ∈ Qp. If √−a ∈ Qp, then Jφ = ∅.

φ has two repelling fixed points x1 =

1 √1−a and x1 = − 1 √1−a ;

(Jφ, φ) ∼ (

2, σ) ;

For x ∈ Qp \ Jφ, limn→∞ φn(x) = ∞.

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• II. φ(x) = ax + 1/x, a ∈ Qp acting on ˆ

Qp – continued

(3) |a|p < 1 : If vp(a) is odd, then Jφ = {0, ∞}. If vp(a) is even, we distinguish two cases : (a) √−a / ∈ Qp ⇒ Jφ = {0, ∞} ; (b) if √−a ∈ Qp,

1

if √a / ∈ Qp, Jφ consists of two parts J1 and J2, with (J1, φ) being topologically conjugate to a subshift of finite type and J2 being the tail of J1.

2

For the case √a / ∈ Qp, there are still lots of work to do.

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