Minimality and chaotic properties of polynomial
dynamical systems on the field of p-adic numbers
Lingmin LIAO (Université Paris-Est Créteil)
Department of Computational & Theoretical Sciences,
International Islamic University Malaysia
Kuantan, August 28th 2014
Lingmin LIAO
University Paris-East Créteil
Minimality and chaotic properties of polynomial p-adic dyn. sys.
1/57
Outline
1
The field Qp of p-adic numbers
2
Dynamical systems and p-adic dynamical systems
3
1-Lipschitz p-adic polynomial dynamical systems
4
p-adic repellers in Qp
Lingmin LIAO
University Paris-East Créteil
Minimality and chaotic properties of polynomial p-adic dyn. sys.
2/57
BOOKS
V. S. Anashin and A. Khrennikov : Applied Algebraic
Dynamics, de Gruyter Exp. Math., vol. 49, Walter de Gruyter &
Co., Berlin, 2009.
W.H. Schikhof : Ultrametric Calculus. An Introduction to
p-Adic Analysis, Cambridge Stud. Adv. Math., vol. 4, Cambridge
University Press, Cambridge, 1984.
F. Q. Gouvêa : p-adic Numbers : An Introduction
(Universitext), Springer, 1997 (2nd edition).
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University Paris-East Créteil
Minimality and chaotic properties of polynomial p-adic dyn. sys.
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The field Qp of p-adic numbers
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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I. The p-adic numbers (K. Hensel, 1897)
p ≥ 2 a prime number.
PN
∀n ∈ N, n = i=0 ai pi (ai = 0, 1, · · · , p − 1)
Ring Zp of p-adic integers :
Zp 3 x =
P∞
i=0
ai pi .
Field Qp of p-adic numbers : fraction field of Zp :
P∞
Qp 3 x = i=v(x) ai pi , (∃v(x) ∈ Z).
Absolute value : |x|p = p−v(x) , metric : d(x, y) = |x − y|p .
3Z3
2 + 3Z3
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1 + 3Z3
Minimality and chaotic properties of polynomial p-adic dyn. sys.
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II. Topology of Qp
p-adic norm of x ∈ Q
|x|p = p−v(x) if x = pv(x)
r
with (r, p) = (s, p) = 1
s
|x|p is a non-Archimidean norm :
| − x|p = |x|p
|xy|p = |x|p |y|p
|x + y|p ≤ max{|x|p , |y|p }
Qp is the | · |p -completion of Q ( Zp = {x ∈ Qp : |x|p ≤ 1} = N)
Development of numbers :
N → Z → Q → R([−1, 1]) → C
N → Z → Q → Qp (Zp ) → Qa.c.
→ Cp
p
Theorem (Ostrowski 1918)
Each non-trivial norm on Q is equivalent to | · | or to | · |p for some p.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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III. Arithmetic in Qp
Addition and multiplication : similar to the decimal way.
”Carrying” from left to right.
Example : x = (p − 1) + (p − 1) × p + (p − 1) × p2 + · · · , then
x + 1 = 0. So,
−1 = (p − 1) + (p − 1) × p + (p − 1) × p2 + · · · .
2x = (p − 2) + (p − 1) × p + (p − 1) × p2 + · · · .
We also have subtraction and division.
Exercise : For p = 2, what is −2 ?
Exercise : For p = 3, write the expansion of 1/5.
Then we can define polynomials and rational maps.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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IV. p-adic analysis VS classical analysis
Something you like :
{an } is Cauchy ⇐⇒ {an } is quasi-Cauchy (an+1 − an → 0).
P∞
n=0 an < ∞ ⇐⇒ an → 0.
p-adic expansion of a p-adic number is unique.
Something you might do not like :
No mean value theorem
f : X ⊂ Qp → Qp , f 0 (x) = 0 ; f = const.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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V. A result of analysis on Qp
f : X(⊂ Qp ) → Qp is continuously differentiable at a ∈ X if the
following exists :
lim
(x,y)→(a,a),x6=y
f (x) − f (y)
.
x−y
Lemma (Local rigidity lemma)
Let U be a clopen set and a ∈ U . Suppose
f : U → Qp is continuously differentiable, f 0 (a) 6= 0.
Then there exists r > 0 such that Br (a) ⊂ U and
|f (x) − f (y)|p = |f 0 (a)|p |x − y|p
(∀x, y ∈ Br (a)).
Exercise : Given a polynomial f , how to find r in the lemma ?
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University Paris-East Créteil
Minimality and chaotic properties of polynomial p-adic dyn. sys.
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Dynamical systems
and p-adic dynamical systems
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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I. Basic notions of dynamical systems
Space : X (topological space, probability space).
Transformation : T (continuous, measurable).
→ Dynamical system (X, T )
Orbit O(x) = {T n x}n∈N ;
T is transitive if O(x) = X (∃x ∈ X) ;
T is minimal if O(x) = X (∀x ∈ X) ;
T is ergodic with respect to an invariant measure µ if
µ(A) = µ(T −1 A) implies µ(A) = 0 or 1 ;
T is uniquely ergodic if ∃ ! invariant probability measure ;
T is strictly ergodic if ”uniquely ergodic” + ”minimal”.
Theorem (Oxtoby 1952) : If the continuous transformation T is
uniquely ergodic (µ is the unique invariant probability measure), then for
any continuous function g : X → R, uniformly, the Birkhoff averages
converge :
Z
n−1
1X
g(T k (x)) → gdµ.
n
k=0
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II. Equicontinuous dynamics
Let (X, d) be a compact metric space. T : X → X is equicontinuous if
∀ > 0, ∃δ > 0 s.t. d(T n x, T n y) < (∀n ≥ 1, ∀d(x, y) < δ).
Theorem (Oxtoby 1952)
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 : 1-Lipschitz transformation is equicontinuous.
Fact 2 : A polynomial f ∈ Zp [x] with coefficients in Zp , which
defines a transformation from Zp to Zp , is 1-Lipschitz, hence is
equicontinuous.
Exercise : Prove Facts 1 and 2.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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III. Topological entropy
Let X be a compact space and T : X → X is continuous. Let α = {Ai }
and β = {Bj } be two (finite) covers of X. The refinement of α and β is
α ∨ β := {Ai ∩ Bj : Ai ∩ Bj 6= ∅}.
entropy of a finite cover α = {A1 , . . . , An } : H(α) := log N (α),
where N (α) is the smallest number of sets of subcovers of α.
entropy of T relative to a cover α :
h(T, α) := lim sup
n→∞
1
−i
H(∨n−1
α).
i=0 T
n
entropy of T :
htop (T ) := sup{h(T, α) : α is a finite cover of X}.
Fact 3 : Topological entropy is a topological conjugate invariant.
Fact 4 : Any equicontinuous (hence 1-Lipschitz) transformation is of
entropy zero.
Exercise : Prove Facts 3 and 4.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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IV. Chaos
Let (X, d) be a compact metric space and T : X → X is continuous.
Chaos of Li–Yorke 1975 : there exists an uncountable subset
S ⊂ X, such that for all x 6= y ∈ S
lim inf d(T n x, T n y) = 0, and lim sup d(T n xT n y) > 0.
n→∞
n→∞
Chaos of Devaney 1986 : transitive, the set of periodic points is
dense and X is infinite.
Huang–Ye 2002 : Devaney ⇒ Li–Yorke.
Blanchard–Glasner–Kolyada–Maass 2002 : entropy> 0 ⇒ Li–Yorke.
• T : X → X is sensitive if
∃ > 0, ∀x, ∀δ > 0, ∃y, ∃n ≥ 0, s. t. d(x, y) < δ, d(T n x, T n y) ≥ .
Exercise : Devaney’s chaos implies sensitivity.
Remark : For interval maps, Devaney ⇔ sensitivity ⇔ ent> 0.
Exercise : Let X = {0, 1}N with a metric d defined by
d(x, y) = 2− min{n≥0:xn 6=yn } . Let T be the left shift :
T (x0 x1 x2 . . . ) = (x1 x2 . . . ). Then (X, T ) is Devaney and ent = log 2.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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V. p-adic polynomial dynamical systems
Let X ⊂ Qp and f ∈ Qp [x] satisfying f : X → X. The couple (X, f ) is a
p-adic polynomial dynamical system.
One is interested in the recurrence property (minimality, chaotic
property, entropy...).
• 1-Lipschitz case (zero entropy) : Let f ∈ Zp [x], and consider the
dynamical system f : Zp → Zp .
Corollary : ”Topological way = measure-theoretic way”
Let X ⊂ Zp be a compact subset. Let f ∈ Zp [x] and f : X → X. Then
the following statements are equivalent :
(1) f is minimal on X. (Every orbit is dense.)
(2) f is ergodic on X with respect to the Haar measure.
(3) f is uniquely ergodic on X.
• Chaotic case (positive entropy). Example : f ∈ Qp [x] and |f 0 (x)|p > 1.
• There are both 1-Lipschitz part and chaotic part.
• More complicated cases.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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VI. 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 sk be the least integer > 1 such that ask ≡ 1 mod pk .
(a) If b 6≡ aj c (mod pk ) for all integers j (0 6 j < sk ), then
pk - (an c − b), for any integer n > 0.
(b) If b ≡ aj c (mod pk ) for some integer j (0 6 j < sk ), then we have
lim
N →+∞
1
1
Card{1 6 n < N : pk | (an c − b)} = .
N
sk
Coelho and Parry 2001 : Ergodicity of p-adic multiplications and the
distribution of Fibonacci numbers.
Exercise : Write the left-side of the formula in (b) of Proposition in form
of Birkhoff averages.
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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VII. 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
Lubin 1994 : iteration of analytic p-adic maps.
Thiran, Verstegen, Weyers 1989 Chaotic p-adic quadratic
polynomials
Mukhamedov, Mendes 2007 : (ax)2 (x + 1)
Mukhamedov, Rozali 2012 : ax(1 − x2 )
Anashin 1994 : 1-Lipschitz transformation (Mahler series)
Yurova 2010 : 1-Lipschitz transformation (Van der Put series)
Coelho, Parry 2001 : ax and distribution of Fibonacci numbers
Gundlach, Khrennikov, Lindahl 2001 : xn
Benedetto, Li Hua-Chieh, Rivera-Letelier, · · · · · ·
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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1-Lipschitz p-adic polynomial
dynamical systems
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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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 2011) minimal decomposition
Let f ∈ Zp [x] with deg f ≥ 2. The space Zp can be decomposed into
three parts :
Zp = P t M t B,
where
P is the finite set consisting of all periodic orbits ;
M := ti∈I Mi (I finite or countable)
→ Mi : finite union of balls,
→ f : Mi → Mi is minimal ;
B is attracted into P t 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/ps Z.
←
Odometer : The transformation τ : x 7→ x + 1 on Z(ps ) .
Theorem (Chabert-Fan-Fares 2009)
Let E be a compact set in Zp 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 (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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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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III. Problems
Problem 1 : Under what condition one has P = B = ∅, and M admits a
unique component ?
(i.e., f : Zp → Zp is minimal ?)
Problem 2 : If (Zp , f ) is not minimal, how to find the complete
decomposition ?
Problem 3 : The solution for deg f = 2 ? (What about the affine
polynomial case ?)
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IV. Minimality on the space Zp
Theorem (Larin 2002 + Knuth 1969), Quadratic polynomials, all p
Let f (x) = ax2 + bx + c with a, b, c ∈ Zp . f is minimal iff
1
a ≡ 0 (mod p), b ≡ 1 (mod p), c 6≡ 0 (mod p), if p ≥ 5,
2
a ≡ 0 (mod 9), b ≡ 1 (mod 3), c 6≡ 0 (mod 3) or
ac ≡ 6 (mod 9), b ≡ 1 (mod 3), c 6≡ 0 (mod 3), if p = 3.
3
a ≡ 0 (mod 2), a + b ≡ 1 (mod 4) and c 6≡ 0 (mod 2), if p = 2.
Theorem (Larin 2002), General polynomials, only for p = 2
P
Let p = 2 and let f (x) = ak xk ∈ 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).
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→ Minimality of general polynomials-Criterion for the case p = 3
P
Let f (x) = ak xk ∈ Z3 [X]. We can suppose a0 = 1 :
a0 ≡ 0 (mod 3) ⇒ f is not minimal.
a0 6≡ 0 (mod 3) ⇒ f is conjugate to a polynomial g with a0 = 1.
Note
A0 =
X
a i , A1 =
i∈2N,i6=0
X
ai , D0 =
i∈1+2N
X
i∈2N,i6=0
iai , D1 =
X
iai .
i∈1+2N
Theorem (Durand-Paccaut 2009), General polynomials, only for p = 3
The system (Z3 , f ) is minimal iff f satisfies A0 ∈ 3Z3 , A1 ∈ 1 + 3Z3 and
one of the following conditions :
P
(1) D0 ≡ 0, D1 ≡ 2, a1 ≡ 1 [3] and A1 + 5 6≡ 0, 3a2 + 3 Pj>0 a5+6j [9] ;
(2) D0 ≡ 0, D1 ≡ 1, a1 ≡ 1 [3] and A0 + 6 6≡ 0, 6a2 + 3 Pj>0 a2+6j [9] ;
(3) D0 ≡ 1, D1 ≡ 0, a1 ≡ 2 [3] and A1 + 5 6≡ 0, 6a2 + 3 Pj>0 a5+6j [9] ;
(4) D0 ≡ 2, D1 ≡ 0, a1 ≡ 2 [3] and A0 + 6 6≡ 0, 3a2 + 3 j>0 a2+6j [9].
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V. Affine polynomials on Zp
Let Ta,b x = ax + b (a, b ∈ Zp ). Denote
U = {z ∈ Zp : |z| = 1}, V = {z ∈ U : ∃m ≥ 1, s.t. z m = 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 `-periodic orbit, with ` the
smallest integer > 1 such that a` = 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} t tn≥1 pn U.
(1) One fixed point {0}.
(2) All (pn U, ax)(n ≥ 0) are conjugate to (U, ax).
`
For (U, Ta,0 ) : pvp (a −1) (p − 1)/` minimal parts, with ` the smallest
integer > 1 such that a` ≡ 1(mod p).
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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Two typical decompositions of Zp
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Minimality and chaotic properties of polynomial p-adic dyn. sys.
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Theorem (Fan-Li-Yao-Zhou 2007) Case p = 2 :
4
a ∈ (U \ V) ∪ {1}, vp (b) < vp (1 − a).
• vp (b) = 0 ⇒ pvp (a+1)−1 minimal parts.
• vp (b) > 0 ⇒ pvp (b) minimal parts.
5
a ∈ U \ V, vp (b) ≥ vp (1 − a)
⇒ (Zp , Ta,b ) is conjugate to (Zp , ax).
Decomposition : Zp = {0} t tn≥1 pn U.
(1) One fixed point {0}.
(2) All (pn U, ax)(n ≥ 0) are conjugate to (U, ax).
2
For (U, Ta,0 ) : 2v2 (a
−1)−2
minimal parts.
Remark : For the case p = 2, all minimal parts (except for the periodic
orbits) are conjugate to (Z2 , x + 1).
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VI. Affine polynomials on Qp
Let ϕ be an affine map defined by
ϕ(x) = ax + b (a, b ∈ Qp , a 6= 0, (a, b) 6= (1, 0)).
If |a| =
6 1 : easy ! For |a| = 1, we have the following conjugacy :
a 6= 1 :
Qp
x−
ax + b -
b
1−a
Qp
x−
?
ax
Qp
b
1−a
?
-
Qp
a=1:
Qp
x+b -
x
b
x
b
?
Qp
Lingmin LIAO
Qp
University Paris-East Créteil
x + 1-
?
Qp
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VII. Affine polynomials on Qp -continued
Theorem (AH. Fan, Y. Fares 2011)
If K = Qp , then
1
ϕ(x) = x + 1 : Qp = Zp ∪
∞
S
n=1
pn U.
Zp is minimal.
pn U contains pn−1 (p − 1) minimal balls with radius 1.
2
ϕ(x) = ax (a is not a root of unity) : Qp = {0} ∪
S
n∈Z
pn U.
0 is fixed.
All subsystems on pn U are conjugate to (U, ϕ).
For (U, ϕ) :
`
(1) Case p ≥ 3 : pvp (a −1) (p − 1)/` minimal balls of same radius,
with ` the smallest integer > 1 such that a` ≡ 1(mod p).
2
(2) Case p = 2 : 2v2 (a −1)−2 minimal balls of same radius.
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Two typical decompositions of Qp
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VIII. Minimal decomp. for quadratic polynomials on Z2
Let
f (x) := Ax2 + Bx + C
(A, B, C ∈ Z2 , A 6= 0).
Note
4 := (B − 1)2 − 4AC.
√
Then f admits fixed points iff 4 ∈ Z2 .
We distinguish three cases :
B ≡ 0 (mod 2)
√
B ≡ 1 (mod 2) and f admits fixed points ( 4 ∈ Z2 )
√
B ≡ 1 (mod 2) and f does not admit any fixed point ( 4 6∈ Z2 )
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Case 1 : B ≡ 0 (mod 2)
There exist α, β, λ ∈ Z2 such that
Z2
Ax2 + Bx-+ C
Z2
αx + β
αx + β
?
Z2
?
- Z2
x2 − λ
Theorem (Fan-L 2011, x2 − λ)
For x2 − λ, λ ∈ Z2 , we have
1
λ ≡ 0 (mod 4) ⇒ two attracting fixed points, basins : 2Z2 , 1 + 2Z2 ;
2
λ ≡ 1 (mod 4) ⇒ one attracting periodic orbit of period 2 ;
3
λ ≡ 2 (mod 4) ⇒ two attracting fixed points, basins : 2Z2 , 1 + 2Z2 ;
4
λ ≡ 3 (mod 4) ⇒ one attracting periodic orbit of period 2.
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Case 2 : B ≡ 1 (mod 2) and f admits fixed points
f admits fixed points, then ∃α, β, b ∈ Z2 , such that
Ax2 + Bx-+ C
Z2
Z2
αx + β
αx + β
?
Z2
?
- Z2
x2 + bx
Theorem (Fan-L 2011)
We obtain the complete decomposition of x2 + bx in Z2 .
(There are countable infinite minimal component.)
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→ Look at a special case.
Theorem (Fan-L 2011, case b = 1)
For (Z2 , x2 + x),
1
The ball 1 + 2Z2 is mapped into 2Z2 .
2
The ball 2Z2 can be decomposed as :


G G
2Z2 = {0} 
2n−1 + 2n Z2  ,
n≥2
and for each n ≥ 2, 2n−1 + 2n Z2 is decomposed as 2n−2 minimal
component :
2n−1 + t2n + 22n−2 Z2 ,
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t = 0, . . . , 2n−2 − 1.
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→ decomposition
of x2 + x
1
0
2
0
8
16
8
8
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University Paris-East Créteil
4
12
24
(mod 24 )
(mod 25 )
24
40
(mod 22 )
(mod 23 )
4
0
0
(mod 2)
0
56
(mod 26 )
Minimality and chaotic properties of polynomial p-adic dyn. sys.
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Case 3 : B ≡ 1 (mod√2) and f has no fixed point
Then ∃α, β, d ∈ Z2 , but,
d 6∈ Z2 , such that
Ax2 + Bx-+ C
Z2
Z2
αx + β
αx + β
?
Z2
?
- Z2
x2 + x − d
Theorem (Fan–L 2011)
√
We obtain the complete decomposition of x2 + x − d ( d 6∈ Z2 ) in Z2 .
→ Look at a special case.
Theorem (Fan-L, 2011, case d ≡ 3 (mod 4) )
√
Let f (x) = x2 + x − d with d 6∈ Z2 and d ≡ 3 (mod 4). Then f |1+2Z2 ,
is minimal and 2Z2 is mapped into 1 + 2Z2 by f .
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IX. Minimal decomp. for Chebyshev polynomials on Z2
For each integer m ≥ 0, the m-th Chebyshev polynomial is defined as
bm/2c
X
m
m − k m−2k−1 m−2k
k
Tm (x) =
(−1)
2
x
.
m−k
k
k=0
Remark : T0 (x) = 1, T1 (x) = 1, Tn+1 (x) = 2xTn (x) − Tn−1 (x).
Theorem (S.L. Fan-L, in preparation)
• If m is even. Then 1 is an attracting fixed point of Tm , and all other
points lie in the attracting basin of 1.
• If m is odd. The dynamics of T1 (x) = x is clear. Write the set of odd
numbers greater than 3 in the following way
{m = 2k + 1 : k ≥ 1, k ∈ Z} =
G
(
G
(2s q ± 1)).
s≥2 q is odd
Then for m ∈
F
(2s q ± 1) for some s ≥ 2. We can decompose Z2 as
q is odd
Z2 = {0, 1, −1 } t (E1
| {z }
G
E2
G
E3 ),
fixed points
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Theorem continued (S.L. Fan–L, in preparation)
where
E1 =
G
G
E1 (n, i),
n≥1 0≤i<2s−1
E2 =
G
G
E2 (n, i),
n≥2 0≤i<2s
E3 =
G
G
E3 (n, i),
n≥2 0≤i<2s
with
E1 (n, i) := 2n (1 + 2i) + 2n+s Z2 (n ≥ 1, 0 ≤ i < 2s−1 ),
E2 (n, i) := 1 + 2n (1 + 2i) + 2n+s+1 Z2 (n ≥ 2, 0 ≤ i < 2s ),
E3 (n, i) := −1 + 2n (1 + 2i) + 2n+s+1 Z2 (n ≥ 2, 0 ≤ i < 2s )
being the minimal components of Tm .
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X. Structure of the square mapping on Z/pZ = Fp
Let f : x 7→ x2 be the square mapping.
Graph G(F∗p ) : vertices are the elements of F∗p and edges are directed
from x to f (x). Let σ(`, k) be the graph consisting a cycle of length `
with a copy of the binary tree Tk of height k attached to each vertex.
Theorem (Rogers 1996)
Let p be an odd prime. Put p = 2k m + 1 where m is odd. Then
G(F∗p ) =
[
d|m
(σ(ordd 2, k) ∪ . . . ∪ σ(ordd 2, k)) .
|
{z
}
ϕ(d)/ordd 2
where ϕ(d) is Euler’s ϕ function, ordd 2 is the order of 2 mod d.
2
4
10
7
1
5
9
8
3
6
Figure : The graphs
G(F∗p ) for
primes p = 11 (k = 1, m = 5 and, d = 1, 5)
[1] V. S. Anashin. Uniformly
distributed sequences of p-adic integers. Diskret. Mat., 14(4):3–64, 2002.
[2] Vladimir Anashin. Ergodic transformations in the space of p-adic integers. In p-adic mathematical physics, volume
826
of AIP Conf. Paris-East
Proc., pages 3–24.
Amer. Inst. Phys.,Minimality
Melville, NY, 2006.
University
Créteil
and chaotic properties of polynomial
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1
16
13
4
9
3
8
14
5
15
12
10
2
7
11
6
Figure : The graphs G(F∗p ) for primes p = 17 (k = 4, m = 1). The vertices
are the elements of F∗p with edges directed from x to x2 .
Remark 1 : 0 is a fixed point in Z/pZ. Further, 0 is a fixed point in Zp
with attracting basin pZp .
Remark 2 : The trees are attracted to the cycles. They lie in attracting
basins of periodic orbits.
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XI. Minimal decomp. for square mapping on Zp
Let C ⊂ Zp \ pZp be the union of balls corresponding to the points in the
cycles.
Theorem (S.L. Fan–L, submitted)
Let p be an odd prime with p = 2k m + 1 where m is an odd integer. Then
C can be decomposed as the union of m periodic points and countably
many minimal components around each periodic orbit.
Let Pm be the set of periodic points, i.e.,
Pm = {x ∈ C : f n (x) = x for some integer n ≥ 1}.
Then Pm ⊂ P and we can decompose Pm in the following way :
G
Pm =
σ̂(ordd 2) t · · · t σ̂(ordd 2)
|
{z
}
d|m
ϕ(d)/ordd 2
where σ̂(`) is a periodic orbit of period `.
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Theorem continued (S.L. Fan–L, submitted)
Let σ̂(`) = (x̂1 , · · · , x̂` ) be one of the periodic orbits of period `. Around
this periodic orbit, we have the following decomposition


G G
G
Sp−n (x̂i ) .
D1 (x̂i ) = {x̂1 , · · · , x̂` } t 
n≥1 1≤i≤`
1≤i≤`
For each n ≥ 1, the set
F
1≤i≤` Sp−n (x̂i ) belongs to the minimal part
p−1
(p−1)·(ordp 2,`)
M and contains
· pvp (2 −1)−1 minimal components, and
ordp 2
`·ordp 2
each minimal component is a union of j := (ordp 2,`)
closed disks of radius
p−1
−n−vp (2
−1)
p
.
F
For each minimal component Mi lying in 1≤i≤` D1 (x̂i ), the subsystem
f : Mi → Mi is conjugate to the adding machine on the odometer Z(ps ) ,
where
(ps ) = (`, `j, `jp, `jp2 , · · · ).
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XII. Ideas and methods-local and global
Origin of the ideas : desJardins and Zieve 1994
→ the study of topological structure of fn : Z/pn Z → Z/pn Z (n ≥ 1)
Local-Global Lemma (Anashin 2005, Chabert, Fan and Fares 2007)
Let X ⊂ Zp be a compact set.
f : X → X is minimal ⇔ fn : X/pn Zp → X/pn Zp for all n ≥ 1.
Predicting the behavior of fn+1 by the structure of fn .
Consider the cycle (x1 , . . . , xk ) in Z/pn Z,
Each xi is lift to be p points {xi + tpn : 0 ≤ t < p} in Z/pn+1 Z.
Linearization : g := f k ,
g(x1 + tpn ) ≡ x1 + (an t + bn )pn (mod pn+1 )
with
an = g 0 (x1 ),
bn =
g(x1 )−x1
.
pn
Linear maps Φ : Φ(t) = an t + bn .
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XIII. Ideas and methods-behavior of cycles
Recall that g = f k : x1 + tpn → x1 + (an t + bn )pn .
So, in the big ball of x1 + pn Zp , the “number t” small ball is sent to
“number an t + bn ” small ball.
Lifts of the cycle (x1 , . . . , xk ) :
Let Xn+1 = {xi + tpn : 0 ≤ t < p}
an ≡ 1, bn 6≡ 0 mod p : fn+1 |Xn+1 has a single cycle of length pk.
We say σ grows.
an ≡ 1, bn ≡ 0 mod p : fn+1 |Xn+1 has p cycles of length k.
We say σ splits.
an ≡ 0 mod p : fn+1 |Xn+1 has a single cycle of length k and the
remaining points of X are mapped into this cycle by f k .
We say σ grows tails.
an 6≡ 0, 1 mod p : fn+1 |Xn+1 has a single cycle of length k and
(p − 1)/d cycles of length kd.
We say σ partially splits.
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Behavior of fn+1
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Case 1
Case 2
Case 3
Case 4
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XIV. Ideas and methods-prediction
Proposition (desJardins-Zieve 1994, Fan–L 2011)
Let n ≥ 1 and σ be a k-cycle of fn and σ̃ be one of its lift. We have
1) if an ≡ 1 (mod p), then an+1 ≡ 1 (mod p) ;
2) if an ≡ 0 (mod p), then an+1 ≡ 0 (mod p) ;
3) if an 6≡ 0, 1 (mod p) and σ̃ is of length k, then an+1 6≡ 0, 1 (mod p) ;
4) if an 6≡ 0, 1 (mod p) and σ̃ is of length k` where ` ≥ 2 is the order of
an in (Z/pZ)∗ , then an+1 ≡ 1 (mod p).
By this result :
1) If σ grows or splits, then any lift σ̃ grows or splits.
2) If σ grows tails, the single lift σ̃ grows tails.
3) If σ partially splits, then the lift σ̃ of the same length as σ partially
splits, and all other lifts of length k` grow or split.
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XV. Ideas and methods-attracting basin
For a cycle σ = (x1 , . . . , xk ) at level n, let
Xi := xi + pn Zp = {x ∈ Zp : x ≡ xi (mod pn )}
(1 ≤ i ≤ k)
be the closed disk of radius p−n corresponding to xi ∈ σ and
Xσ :=
k
G
Xi
i=1
be the compact-open set corresponding to the cycle σ.
Proposition (Fan–L 2011)
If σ = (x1 , · · · , xk ) is a growing tails cycle at level n, then f has a kperiodic orbit in the compact-open set Xσ , and with Xσ as its attracting
basin.
P roof : Notice that If σ grows tails, the single lift σ̃ also grows tails. So
for all levels, all points are attracted to the single cycle.
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XVI. Ideas and methods-minimal parts
Lemma (desJardins-Zieve 1994, Fan–L 2011)
Let p ≥ 3 be a prime and n ≥ 2 be an integer. If σ is a growing cycle of
fn and σ̃ is the unique lift of σ, then σ̃ grows.
Proposition (Fan–L 2011)
Let p ≥ 3 be a prime and n ≥ 2 be an integer. If σ is a growing cycle of
fn , then σ produces a minimal component, i.e., the set Xσ is a minimal
subsystem of f .
P roof : By the above lemma, if σ̃ is the lift of σ, then σ̃ also grows.
Applying the lemma gain, if σ̃ grows, then the lift of σ̃ grows.
Consecutively, we find that the descendants of σ will keep on growing.
(In this case, we usually say σ always grows or grows forever.) Hence, fm
is minimal on Xσ /pm Zp for each m ≥ n. Therefore, by Local-Global
Lemma, (Xσ , f ) is minimal.
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p-adic repellers in Qp
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I. Settings
f : X → Qp , X ⊂ Qp compact open.
Assume that
1
2
f −1 (X)
F ⊂X;
X = i∈I Bp−τ (ci ) (with some τ ∈ Z), ∀i ∈ I, ∃τi ∈ Z s.t.
|f (x) − f (y)|p = pτi |x − y|p
Define Julia set :
∞
\
Jf =
(∀x, y ∈ Bp−τ (ci )).
(1)
f −n (X).
n=0
We have f (Jf ) ⊂ Jf . (X, Jf , f ) is called
→ a p-adic weak repeller if all τi ≥ 0 in (1), but at least one > 0.
→ a p-adic repeller if all τi > 0 in (1).
Theorem (Fan-L-Wang-Zhou, 2007)
Let f ∈ Qp [x] be a polynomial with f 0 (x) 6= 0 for all x ∈ Qp . Then there
exist a compact-open set X and an integer τ such that the above two
conditions are satisfied. Further, for all x 6∈ Jf , |f n (x)|p → ∞.
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For any i ∈ I, let
Ii := {j ∈ I : Bj ∩ f (Bi ) 6= ∅} = {j ∈ I : Bj ⊂ f (Bi )}.
Define A = (Ai,j )I×I :
Aij = 1 if j ∈ Ii ;
Aij = 0 otherwise.
If A is irreducible, we say that (X, Jf , f ) is transitive.
(ΣA , σ) be the corresponding subshift.
For i, j ∈ I, i 6= j, let κ(i, j) be the integer s.t. |ci − cj |p = p−κ(i,j) .
for x = (x0 , x1 , · · · , xn , · · · ), y = (y0 , y1 , · · · , yn , · · · ) ∈ ΣA , define
df (x, y)
df (x, y)
= p−τx0 −τx1 −···−τxn−1 −κ(xn ,yn )
= p
−κ(x0 ,y0 )
(if n 6= 0),
(if n = 0)
where n = n(x, y) = min{i ≥ 0 : xi 6= yi }.
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II. Results
Theorem (Fan-L-Wang-Zhou, 2007)
Let (X, Jf , f ) be a transitive p-adic weak repeller with matrix A. Then
the dynamics (Jf , f, | · |p ) is isometrically conjugate to the shift dynamics
(ΣA , σ, df ).
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III. Examples-Full shifts
→ Example 1
Let c = pcτ0 ∈ Qp with |c0 |p = 1 and τ ≥ 1.
Define p-adic logistic map fc : Qp −→ Qp
fc (x) = cx(x − 1).
X = pτ Zp t (1 + pτ Zp ),
T∞
Jc = n=0 fc−n (X).
Theorem (Fan-L-Wang-Zhou, 2007)
(Jc , fc ) is conjugate to ({0, 1}N , σ).
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→ Example 2
Let a ∈ Zp , a ≡ 1 (mod p) and m ≥ 1 be an integer.
Consider fm,a : Qp −→ Qp :
fm,a (x) =
xp − ax
.
pm
Im,a = {0 ≤ k < pm : k p − ak ≡ 0 (mod pm )}
T∞
F
−n
(X).
Xm,a = k∈Im,a (k + pm Zp ), J = n=0 fm
Theorem (Fan-L-Wang-Zhou, 2007)
(J, fm,a ) is conjugate to ({0, . . . , p − 1}N , σ).
Theorem (Woodcock and Smart 1998)
(J, f1,1 ) is conjugate to ({0, . . . , p − 1}N , σ).
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IV. Cubic polynomials
f (x) = x3 + ax with |a|p > 1
(Possible subsystem : full shift with two symbols)
f (x) = x3 + a where |a|p > 1
(Possible subsystem : full shift with two symbols)
f (x) = x3 + ax2 where |a|p > 1 (Mukhamedov and Mendes
2007)
(Possible subsystems : minimal components, full shift with two
symbols).
on Q2 (Fan-L-Wang-Zhou 2007).
Consider f (x) = x(x−1)(x+1)
2
Exercise : |f (x) − f (y)|p = |f 0 (y)|p |x − y|p ∀x, y ∈ Z2 , |x − y|p ≤ 1/4.
F3
Take X = k=0 (k + 4Z2 ). Then τ0 = τ2 = 2 and τ1 = τ3 = 1, and

1
 1

A=
0
1
0
0
1
0
1
0
0
0

0
0 

1 
0
The topological entropy of (Jf , f ) is equal to log 1.6956... where
1.6956... is the maximal eigenvalue of A.
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Three special examples :
f (x) = 12 x3 + 2x (attracting points : 0 and ∞),
f (x) =
x3
6
+ x2 −
x
2
on Z2 , attracting and expanding mixed :
A := 1 + 4 + 8Z2 2 + 4 + 16Z2 =: B
C := 1 + 8Z2 2 + 4 + 8 + 16Z2 =: D.
2
+ 1. (Rivera-Letelier 2005) : the critical point 0 is
f (x) = x (x−1)
2
sent to the repelling fixed point 1. On the Julia set, it is not a
subshift of finite type.
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V. Polynomials with higher orders
Let m < τ ∈ N∗ be two positive integers and c0 , x0 , . . . , xm ∈ Zp .
Suppose vp (c0 ) = 0. Consider
f (x) =
c0 (x − x0 )(x − x1 ) · · · (x − xm )
.
pτ
Assume that
Y
(xi − xj ) 6≡ 0 (mod p).
j6=i
−n
Let X = ∪i (xi + pτ Zp ) and define Jf = ∩∞
(X).
n=0 f
Theorem (L, unpublished)
We have x ∈
/ Jf , |f n (x)|p → ∞ and (Jf , f ) is conjugate to the full shift
(Σm , σ) with m symbols.
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VI. Polynomials with higher orders-continued
Let m ∈ N∗ be a positive integer and x1 , . . . , xm ∈ Zp .
Suppose xi ≡ x0 6≡ 0 (mod p)(1 ≤ i < p) and xi 6≡ xj (mod p2 ) (i 6= j).
2 )···(x−xm )
.
Consider f (x) = x(x−x1 )(x−x
pm−1
Take
X = pm Zp ∪ (kpm−1 + pm Zp ) ∪
m
[
−n
xi + pm Zp , Jf := ∩∞
(X).
n=0 f
j=1
Theorem (L, unpublished)
System (Jf , f ) is conjugate to a subshift of finite

1
1
0
···
 0
0
1
···

 1
0
0
···
A=
 1
0
0
···


···
···
1
0
0
···
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type of m + 2 symbols.

0
1 

0 

0 


0
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