Diophantine approximation with applications to dynamical systems by Michel Waldschmidt Abstract ...

18 Pages·2014·0.31 MB·English

Update: 25/01/2014
Papua New Guinea PNG University of Technology,
International Conference on Pure and Applied Mathematics
ICPAM–LAE 2013, November 26 – 28, 2013
Diophantine approximation
with applications to dynamical systems
by
Michel Waldschmidt
Abstract
Dynamical systems were studied by Henri Poincar´e and Carl Ludwig Siegel,
who developed the theory of celestial mechanics. The behavior of a holo-
morphicdynamicalsystemnearafixedpointdependsonaDiophantinecon-
dition, already introduced by Joseph Liouville in 1844 when he constructed
the first examples of transcendental numbers. One of the deepest results in
Diophantine approximation is the Subspace Theorem of Wolfgang Schmidt.
We give an application related with linear recurrence sequences and expo-
nential polynomials, involving a dynamical system on a finite dimensional
vector space.
Classification: primary 11J87, secondary 32H50 37F50
Keywords: Dynamical systems; Diophantine approximation; holomorphic dynam-
ics; Thue–Siegel–Roth Theorem; Schmidt’s Subspace Theorem; S-unit equations;
linear recurrence sequences; exponential polynomials; Skolem–Mahler–Lech Theo-
rem.
Acknowledgments
I was invited in January 2010 by Ashok Agrawal to participate to the 97th Indian
ScienceCongressinThiruvananthapuram(Kerala). Duringthismeeting, following
the suggestion of Jugal K. Verma, A. J. Parameswaran, then Director of the Ker-
ala School of Mathematical Science (KSOM) in Kozikhode, invited me to visit his
institute. During my visit just after the congress, we discussed some work by S.G.
1
Dani involving a question relating dynamical systems with Diophantine approxi-
mation (see Corollary 7.4 below). I had a correspondence with Pietro Corvaja and
Umberto Zannier on this topic at that time.
AfterhebecametheDirectorofKSOM,M.Manickamsuggestedmetoorganize
a workshop in his institute. With Yann Bugeaud, S.G. Dani and Pietro Corvaja,
we selected the topic number theory and dynamical systems. This workshop took
place in February 2013.
ThistextisareportonthelectureIgaveinLae,attheInternationalConference
on Pure and Applied Mathematics ICPAM–LAE 2013, The Papua New Guinea
(PNG) University of Technology (November 26 - 28, 2013), where I was invited by
Kenneth Nwabueze.
I am pleased to express my thanks to Ashok Agrawal, Jugal K. Verma, A. J.
Parameswaran, Yann Bugeaud, S.G. Dani, Pietro Corvaja, Umberto Zannier, M.
Manickam and Kenneth Nwabueze.
IwishalsotothankAjayaSinghandJorgeJimenezUrroz: whenwedidatrek
together around the Annapurna in fall 2011 and to the Everest Base Camp in fall
2012, we carried with us some papers on dynamical systems ([21] in 2011, [2] and
[10] in 2012) where I learned the basis facts on this topic.
1 Iteration of a map
Consider a set X and map f : X → X. We denote by f2 the composition
map f ◦f : X → X. More generally, we define inductively fn : X → X by
fn = fn−1 ◦f for n ≥ 1, with f0 being the identity. The orbit of a point
x ∈ X is the set
{x,f(x),f2(x),...} ⊂ X.
A fixed point is an element x ∈ X such that f(x) = x. Hence, a fixed point
is a point x, the orbit of which has only the element x.
A periodic point is an element x ∈ X for which there exists n ≥ 1 with
fn(x) = x. The smallest such n is the length of the period of x, and all such
n are the multiples of the period length. The orbit
{x,f(x),...,fn−1(x)}
has n elements. For instance, a fixed point is a periodic point of period
length 1.
2 Endomorphisms of a vector space
Take for X a finite dimensional vector space V over a field K and for f :
V → V a linear map. A fixed point of f is nothing else than an eigenvector
2
with eigenvalue 1. A periodic point of f is an element x ∈ V such that there
exists n ≥ 1 with fn(x) = x, hence, f has an eigenvalue λ with λn = 1 (λ is
a root of unity).
IfV hasdimensiondandifwechooseabasisofV,thentof isassociated
a d×d matrix A with coefficients in K. Then, for n ≥ 1, fn is the linear
map associated with the matrix An. To compute An, we write the matrix
A as a conjugate to either a diagonal or a Jordan matrix
A = P−1DP,
where P is a regular d×d matrix. Then, for n ≥ 0,
An = P−1DnP.
If D is a diagonal matrix with diagonal (λ ,...,λ ), then Dn is a diagonal
1 d
matrix with diagonal (λn,...,λn), so that
1 d
λn ··· 0 
1
An = P−1 ... ... ... P.
0 ··· λn
d
Two examples are given in the Appendix.
In general, the matrix A can be written A = P−1DP with diagonal blocs
 
D ··· 0
1
D =  ... ... ... 
0 ··· D
k
where, for i = 1,...,k, D is a d ×d Jordan matrix
i i i
 
λ 1 0 ··· 0
i
0 λi 1 ··· 0
Di =  ... ... ... ... ... 
 
0 0 0 ··· 1
0 0 0 ··· λ
i
with d +··· +d = d (the diagonal case is the case d = ··· = d = 1,
1 k 1 k
k = d). Then, for n ≥ 1,
Dn 0 
1
Dn =  ... 
 
0 Dn
k
3
with
λn nλn−1 (cid:0)n(cid:1)λn−2 ··· (cid:0) n (cid:1)λn−di−2 (cid:0) n (cid:1)λn−di−1
 0i λin n2λni−1 ··· (cid:0)din−2(cid:1)λni−di−3 (cid:0)din−1(cid:1)λni−di−2
Dn =  ... ...i .i.. di−3 ...i di−2 ...i .
i  0 0 0 ··· nλn−1 (cid:0)n(cid:1)λn−2 
 i 2 i 
 0 0 0 ··· λn nλn−1 
 i i 
0 0 0 ··· 0 λn
i
3 Holomorphic dynamic
Our second and main example of a dynamical system is with an open set V
in C and an analytic (=holomorphic) map f : V → V. The main goal will
be to investigate the behavior of f near a fixed point z ∈ V. So we assume
0
f(z ) = z . The local behavior of the dynamics defined by f depends on the
0 0
derivative f(cid:48)(z ) of f at the fixed point:
0
• If f(cid:48)(z ) = 0, then z is a super–attracting point.
0 0
• If 0 < |f(cid:48)(z )| < 1, then z is an attracting point.
0 0
• If |f(cid:48)(z )| > 1, then z is a repelling point.
0 0
• If |f(cid:48)(z )| = 1, then z is an indifferent point.
0 0
When |f(cid:48)(z )| = 1, the point z is a rationally indifferent point or a
0 0
parabolic point if f(cid:48)(z ) is a root of unity and is an irrationally indifferent
0
point if f(cid:48)(z ) is not a root of unity ([2], § 6.1, [10] §2.2).
0
We wish to mimic the situation of an endomorphism of a vector space:
in place of a regular matrix P, we introduce a local change of coordinates
h. Let D be the open unit disc in C and g : D → D an analytic map with
g(0) = 0. We say that f and g are conjugate if there exists an analytic map
h : V → D such that h(z ) = 0, h(cid:48)(z ) (cid:54)= 0 and h◦f = g◦h:
0 0
f
z ∈ V −−−→ V (cid:51) z f(z ) = z
0 0 0 0
 
 
h  h
(cid:121) (cid:121)
0 ∈ D −−−→ D (cid:51) 0 g(0) = 0
g
Assume f : V → V and g : D → D are conjugate: h◦f = g ◦h. Then we
have
h◦f2 = h◦f ◦f = g◦h◦f = g◦g◦h = g2◦h
and by induction h◦fn = gn◦h for all n ≥ 0.
An important special case is when g is a homothety: g(z) = λz.
4
Lemma 3.1. Assume f : V → V is conjugate to the homothety g(z) = λz.
Then
(a) λ = f(cid:48)(z ).
0
(b) Il λ is not a root of unity, then there exists a unique h : D → D with
h(cid:48)(z ) = 1 and h◦f = g◦h.
0
Hence,inthiscase,f isconjugatetoitslinearpartz → z +(z−z )f(cid:48)(z ).
0 0 0
One says that f is linearizable.
Proof. For part (a), take the derivative of h◦f = g◦h at z :
0
h(cid:48)(z )f(cid:48)(z ) = λh(cid:48)(z )
0 0 0
and use h(cid:48)(z ) (cid:54)= 0.
0
For part (b), the unicity when z is not a rationally indifferent point,
0
follows by induction from the equality between the Taylor expansions of
h◦f and g◦h at z = z .
0
Defineλ = f(cid:48)(z ). ThefollowingresultisduetoG.KœnigsandH.Poin-
0
car´e(1884)—seeforinstance,[2],§6.3,[10]Th. 2.2,[13]§1,[14]§6. Several
proofs are given in [2], § 6.3.
Theorem 3.2 (Kœnigs–Poincar´e). Assume λ (cid:54)= 0 and |λ| (cid:54)= 1. Then f is
linearizable.
When λ = 0, f has a zero of multiplicity n ≥ 2 at z and is conjugate to
0
z (cid:55)→ zn (A. B¨ottcher) - see [14] Th. 6.7.
Assume now |λ| = 1. Write λ = e2iπθ. The real number θ is the rotation
number of f at z . It was conjectured in 1912 by E. Kasner that f is always
0
linearizable, meaning that f is conjugate to the rotation z (cid:55)→ e2iπθz. In
1917, G.A. Pfeiffer produced a counterexample. In 1927, H. Cremer proved
that in the generic case, f is not linearizable. In 1942, C.L. Siegel proved
that if θ satisfies a Diophantine condition (see §4), then f is linearizable. In
1965, A.D.BrjunorelaxedSiegel’sassumption. In1988, J.C.Yoccozshowed
that if θ does not satisfies Brjuno’s condition, then the dynamic associated
with
f(z) = λz+z2
has infinitely many periodic points in any neighborhood of 0, hence, is not
linearizable. See [10] §2.2, [13] §3 and [14] §8.
5
4 Diophantine condition
Siegel’sDiophantineconditionontherotationnumberθisthatnogoodratio-
nal approximation p/q of θ can have a small denominator q. The same con-
dition was introduced earlier by Liouville, who proved in 1844 that Siegel’s
Diophantine condition is satisfied if θ is an algebraic number.
Recall that a complex number α is algebraic if there exists a nonzero
polynomial f ∈ Z[X] such that f(α) = 0. The smallest degree of such
√
a polynomial is the degree of the algebraic number α. For instance, 2,
√ √
i = −1, 3 2, e2iπa/b (for a and b integers, b > 0) are algebraic numbers.
ThereexistquinticpolynomialsX5+aX+bwithaandbinZhavingGalois
group the symmetric group S or the alternating group A which are not
5 5
solvable, their roots are algebraic numbers but cannot be expressed using
radicals.
A number which is not algebraic is transcendental. The existence of
transcendental numbers was not known before 1844, when Liouville pro-
duced the first examples, like
(cid:88) 1
ξ = ·
10n!
n≥0
The idea of Liouville is to prove a Diophantine property of algebraic num-
bers, namelythatrationalnumberswithsmalldenominatorsdonotproduce
sharp approximations. Hence, a real number with too good rational approx-
imations cannot be algebraic. For instance, with the above number ξ and
q = 10N!,
N
(cid:88) p 2 2
p = 10N!−n!, 0 < ξ− < = ·
q 10(N+1)! qN+1
n=0
Theorem 4.1 (Liouville’s inequality, 1844). Let α be an algebraic number
of degree d ≥ 2. There exists c(α) > 0 such that, for any p/q ∈ Q with
q > 0,
(cid:12) (cid:12)
(cid:12) p(cid:12) c(α)
(cid:12)α− (cid:12) > ·
(cid:12) q(cid:12) qd
A real number θ satisfies a Diophantine condition if there exists a con-
stant κ > 0 such that
(cid:12) (cid:12)
(cid:12) p(cid:12) 1
(cid:12)θ− (cid:12) >
(cid:12) q(cid:12) qκ
for all p/q ∈ Q with q > 1.
6
An irrational real number is a Liouville number if it does not satisfy a
Diophantine condition.
In dynamical systems, a property is satisfied for a generic rotation num-
ber θ if it is true for all real numbers in a countable intersection of dense
open sets — these sets are called G sets by Baire, who calls meager the
δ
complement of a G set. According to Baire’s Theorem, a G set is dense
δ δ
in R.
The set of numbers which do not satisfy a Diophantine condition is a
generic set. However, for Lebesgue measure, the set of Liouville numbers
(i.e. the set of numbers which do not satisfy a Diophantine condition) has
measure zero.
In terms of continued fraction (see [10] §2.2, [13] §4), the Diophantine
condition (of Liouville and Siegel) can be written
logq
n+1
sup < ∞,
logq
n≥1 n
while the condition of Brjuno is
(cid:88) logqn+1
< ∞.
q
n
n≥1
If a number θ satisfies the Diophantine condition, then it satisfies Brjuno’s
condition. However,thereare(transcendental)numberswhichdonotsatisfy
the Diophantine condition, but satisfy Brjuno’s condition.
5 Schmidt’s Subspace Theorem
In the lower bound
(cid:12) (cid:12)
(cid:12) p(cid:12) c(α)
(cid:12)α− (cid:12) >
(cid:12) q(cid:12) qd
for α real algebraic number of degree d ≥ 3, the exponent d of q in the
denominator of the right hand side was replaced by κ with
• any κ > (d/2)+1 by A. Thue (1909),
√
• any κ > 2 d by C.L. Siegel in 1921,
√
• any κ > 2d by F.J. Dyson and A.O. Gel’fond in 1947,
• any κ > 2 by K.F. Roth in 1955.
Theorem 5.1 (Thue–Siegel–RothTheorem). For any real algebraic number
α, for any (cid:15) > 0, the set of p/q ∈ Q with |α−p/q| < q−2−(cid:15) is finite.
7
An equivalent statement is:
For any real algebraic number α and for any (cid:15) > 0, the set of
p/q ∈ Q such that
q|qα−p| < q−(cid:15)
is finite.
The conclusion can be phrased:
For any real algebraic number α and for any (cid:15) > 0, the set of
(p,q) ∈ Z2 such that
q|qα−p| < q−(cid:15)
is contained in the union of finitely many lines in Z2.
A powerful generalization has been achieved in 1970 by W.M. Schmidt.
Here is a special case of his Subspace Theorem [5, 6, 9, 12, 17, 24, 25].
Theorem 5.2 (Schmidt’sSubspaceTheorem). Let m ≥ 2 be an integer and
L ,...,L be m independent linear forms in m variables with algebraic
0 m−1
coefficients. Let (cid:15) > 0. Then the set
(cid:8)x = (x ,...,x ) ∈ Zm ; |L (x)···L (x)| ≤ |x|−(cid:15)(cid:9)
0 m−1 0 m−1
is contained in the union of finitely many proper subspaces of Qm.
Example: For m = 2, L (x ,x ) = x , L (x ,x ) = αx −x , we recover
0 0 1 0 1 0 1 0 1
Roth’s Theorem.
6 Generalized S–unit equation
TheproofofSchmidt’sSubspaceTheoremhasanarithmeticnature; thefact
that the linear forms have algebraic coefficients is crucial. The conclusion
does not hold without this assumption.
However, there are specializations arguments1 ([16] §4, [18] §2, [19] §9)
which enable one to deduce consequences without any arithmetic assump-
tion: these corollaries are valid for fields of zero characteristic in general.
1As pointed out to me by Umberto Zannier, there are also independent (and easier)
arguments based on derivations which give Theorem 6.1 in the ”transcendental case”
(reducingittothealgebraiccaseorprovingitcompletely,dependingontheassumptions).
8
An example is the so–called Theorem of the generalized S–unit equation,
achievedinthe1980’sbyJ.H.Evertse,A.J.vanderPoortenandH.P.Schlick-
ewei. It relies on a generalization of Schmidt’s Subspace Theorem which
rests on works by Schmidt, Schlickewei and others, involving p–adic num-
bers [6, 12, 24].
Theorem 6.1 (Evertse, van der Poorten, Schlickewei). Let K be a field of
characteristic zero, let G be a finitely generated multiplicative subgroup of
the multiplicative group K× = K \{0} and let n ≥ 2. Then the equation
u +u +···+u = 1,
1 2 n
where the unknowns u ,u ,··· ,u take their values in G, for which no non-
1 2 n
trivial subsum
(cid:88)
u ∅ =(cid:54) I ⊂ {1,...,n}
i
i∈I
vanishes, has only finitely many solutions.
7 Linear recurrence sequences and exponentials
polynomials
Let K be a field of zero characteristic. A sequence (u ) of elements of K
n n≥0
is a linear recurrence sequence if there exist an integer d ≥ 1 and elements
a ,a ,...,a of K with a (cid:54)= 0 such that, for n ≥ 0,
0 1 d−1 0
u = a u +···+a u +a u . (7.1)
n+d d−1 n+d−1 1 n+1 0 n
In matrix notation, (7.1) can be written U = AU , with
n+1 n
   
0 1 0 ··· 0 u
n
0 0 1 ··· 0   un+1 
A =  ... ... ... ... ...  and Un =  ... .
   
0 0 0 ··· 1  un+d−2
a a a ··· a u
0 1 2 d−1 n+d−1
Hence U = AnU for all n ≥ 0. Such a sequence (u ) is determined
n 0 n n≥0
by the coefficients a ,a ,...,a and by the initial values u ,u ,...,u .
0 1 d−1 0 1 d−1
Given a = (a ,a ,...,a ) ∈ Kd, the set of sequences (u ) of elements
0 1 d−1 n n≥0
of K satisfying (7.1) is a K vector space V of dimension d. A basis of V is
a a
obtained by taking for (u ,u ,...,u ) the elements of a basis of Kd.
0 1 d−1
9
Let
det(XI −A) = Xd−a Xd−1−···−a X −a
d d−1 1 0
bethecharacteristicpolynomialofA. Denotebyα ,...,α itsdistinctroots
1 k
and by s ,...,s their multiplicities, so that
1 k
k
(cid:89)
Xd−a Xd−1−···−a X −a = (X −α )si.
d−1 1 0 i
i=1
ComputingAn asmentionedin§2, onededuces thatthereexistpolynomials
A ,...,A with A of degree < s such that
1 k i i
k
(cid:88)
u = A (n)αn. (7.2)
n i i
i=1
In other terms, the d sequences (njαn) , 0 ≤ j < d , 1 ≤ i ≤ k consti-
i n≥0 i
tute a basis for V . Equation (7.2) shows that a linear recurrence sequence
a
is given by an exponential polynomial. Conversely, a sequence given by an
exponential polynomial (7.2) is a linear recurrence sequence. Another char-
acterization is that (u ) is a linear recurrence sequence if and only if the
n n≥0
generating series
u +u z+···+u zn+···
0 1 n
is the Taylor series of a rational fraction A(z)/B(z), where the degree of the
denominator B is larger than the degree of the numerator A. Dropping this
condition on the degrees amounts to asking that there exists n ≥ 0 such
0
that the sequence (u ) is a linear recurrence sequence.
n−n0 n≥0
Theorem 6.1 on the generalized S–unit equation, applied to the mul-
tiplicative subgroup of K× generated by α ,...,α , yields the following
1 k
theorem — see [7, 11, 15, 19, 20, 23, 27]:
Theorem 7.3 (Skolem–Mahler–Lech). Given a linear recurrence sequence
(u ) , the set of indices n ≥ 0 such that u = 0 is a finite union of
n n≥0 n
arithmetic progressions.
An arithmetic progression is a set of positive integers of the form
{n ,n +r,n +2r,...}.
0 0 0
Here, we allow r = 0, which means that we consider a single point as an
arithmetic progression of ratio 0.
TheoriginalproofsofTheorem7.3didnotusetheargumentsinvolvedin
the proof of Theorem 6.1, but were more elementary. T.A. Skolem treated
10