Dangbo D´ecembre 2016 Abstract Institut de Math´ematiqueset de Sciences Physiques (IMSP) http://imsp-benin.com/home/ Centre d’Excellence Africain en Sciences Math´ematiques et Applications Dynamical systems were studied by Henri Poincar´e and Carl http://imsp-benin.com/ceasma Ludwig Siegel, who developed the theory of celestial mechanics. The behavior of a holomorphic dynamical system near a fixed point depends on a Diophantine condition. Diophantine approximation with
applications to dynamical systems Along these lines, we give a survey on Diophantine approximation, culminating with the subspace theorem of Wolfgang Schmidt. Michel Waldschmidt Universit´ePierre et Marie Curie (Paris 6) France http://www.imj-prg.fr/~michel.waldschmidt/
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Celestial mechanics Edward Norton Lorenz (1917 – 2008)
Classical mechanics : Sir Isaac Newton (1643 – 1727)
In chaos theory, the butterfly e↵ect is the sensitive dependency The solar system on initial conditions in which a small change at one place in a • deterministic nonlinear system can result in large di↵erences in The three body problem alaterstate.Thenameofthee↵ect, coined by Edward • Lorenz, is derived from the theoretical example of the formation of a hurricane being contingent on whether or not a Two body problem • distant butterfly had flapped its wings several weeks earlier.
3/55 4/55 Lorenz’s butterfly e↵ect Henri Poincar´e (1854 – 1912)
Two states di↵ering by imperceptible amounts may eventually evolve into two considerably di↵erent states. If, then, there is any error whatever in observing the present state — and in any real system such errors seem inevitable — an acceptable prediction of an instantaneous state in the distant future may well be impossible. In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting seems to be nonexistent.
Lorenz’s description of the butterfly e↵ect followed in 1969. Credit Photos: However, recent research shows that complex systems may not http://www-history.mcs.st-andrews.ac.uk/history behave like systems with fewer parameters.
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Carl Ludwig Siegel (1896 – 1981) Dynamical System : iteration of a map
Consider a set X and map f : X X. We denote by f 2 the composed map f f : X X. ! !
More generally, we define inductively f n : X X by n n 1 0 ! f = f f for n 1, with f being the identity.
The orbit of a point x X is the sequence 2 (x, f(x),f2(x),...)
of elements of X.
7/55 8/55 Fixed points, periodic points Example 1 : endomorphism of a vector space
A fixed point is an element x X such that f(x)=x. A fixed 2 Take for X afinitedimensionalvectorspaceV over a field K point is a point, the orbit of which has one element x. and for f : V V alinearmap. ! A periodic point is an element x X for which there exists A fixed point of f is an element x V such that f(x)=x, n 1 with f n(x)=x. The smallest2 such n is the length of hence, it is nothing else than an eigenvector2 with eigenvalue 1. the period of x, and all such n are multiples of the period length. The orbit A periodic point of f is an element x V such that there n 1 n 2 x, f(x),...,f (x) exists n 1 with f (x)=x, hence, f has an eigenvalue { } with n =1 (root of unity). has n elements. If V has dimension d and if we choose a basis of V , then to f For instance, a fixed point is a periodic point of period length is associated a d d matrix A with coe cients in K. 1. ⇥
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Associated matrix Diagonal form
n When f is the linear map associated with the d d matrix A, If D is diagonal with diagonal ( 1,..., d), then D is n ⇥ n n then, for n 1, f is the linear map associated with the diagonal with diagonal ( 1 ,..., d ) and matrix An. n 1 0 n 1 . A = P . P. To compute An, we write the matrix A as a conjugate to 0 . 1 0 n either a diagonal or a Jordan matrix B d C @ A 1 A = P DP, Exercise : compute An for n 0 and for each of the two where P is a regular d d matrix. Then, for n 0, ⇥ matrices 10 01 n 1 n and . A = P D P. 12 11 ✓ ◆ ✓ ◆
11 / 55 12 / 55 Camille Jordan (1838 – 1922) Example 2 : holomorphic dynamic
If A cannot be diagonalized, it Our second and main example of a dynamical system is with can be put in Jordan form an open set in C and an analytic (=holomorphic) map with diagonal blocs like f : . TheV main goal will be to investigate the behavior V ! V 10 0 of f near a fixed point z0 . So we assume f(z0)=z0. 0 1 ··· 0 2 V 0 . . . ···. . 1 ...... The local behavior of the dynamics defined by f depends on B C B000 1C the derivative f 0(z0) of f at the fixed point. B ··· C B000 C B ··· C @ A If f 0(z0) < 1, then z0 is an attracting point. For instance, for d =2, | |
1 1 If f 0(z ) > 1, then z is a repelling point. A = P DP with D = , 0 0 0 | | ✓ ◆ and The most interesting case is f 0(z0) =1 n n 1 | | n 1 n A = P P. 0 n ✓ ◆ 13 / 55 14 / 55
Conjugate holomorphic maps Local behavior
We wish to mimic the situation of an endomorphism of a Assume f : and g : are conjugate : there exists vector space : in place of a regular matrix P , we introduce a V ! V D ! D h : , with h0(z ) =0and h f = g h. local change of coordinates. Let be the open unit disc in C D ! D 0 6 and g : an analytic mapD with g(0) = 0. We say that f D ! D and g are conjugate if there exists an analytic map h : , From h0(z ) =0, one deduces that h is unique up to a V ! D 0 6 with h0(z0) =0, such that h(z0)=0and h f = g h. multiplicative nonzero factor. 6
f z z f(z )=z Further, 0 2 V ! V 3 0 0 0 h h h f 2 = h f f = g h f = g g h = g2 h 0 ? ? 0 g(0) = 0 2?D !g D?3 y y and, by induction, h f n = gn h for all n 0.
15 / 55 16 / 55 Linearization of germs of analytic di↵eomorphisms Johann Samuel K¨onig (1712 – 1757) of one complex variable
Define = f 0(z0). Lemma. If f is conjugate to the homothety g(z)= z, then Theorem (K¨onigs and = f (z ). 0 0 Poincar´e). For 0, 1 , Hence, in this case, f is conjugate to its linear part. One says f is linearizable.| | 62 { } that f is linearizable.
Proof. Take the derivative of h f = g h at z0 : For =0and z =0, f has a zero of multiplicity n 2 at 0 0 and is conjugate to z zn (A. B¨ottcher). h0(z0)f 0(z0)= h0(z0) 7! We are interested in the case =1. It was conjectured in | | and use h0(z0) =0. 1912 by E. Kasner that f is always linearizable. In 1917, G.A. 6 Pfei↵er produced a counterexample. In 1927, H. Cremer proved that in the generic case, f is not linearizable.
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The case =1 C.L. Siegel, A.D. Brjuno, J.C. Yoccoz | | Assume =1. Write = e2i⇡✓. The real number ✓ is the | | Carl Ludwig Siegel Jean-Christophe Yoccoz rotation number of f at z0. (1896 – 1981) (1957 — 2016) Alexander Dmitrijewitsch Brjuno In 1942, C.L. Siegel proved that if ✓ satisfies a Diophantine (1940 – ) condition, then f is conjugate to the rotation z e2i⇡✓z. 7! In 1965, A.D. Brjuno relaxed Siegel’s assumption.
In 1988, J.C. Yoccoz showed that if ✓ does not satisfies Brjuno’s condition, then the dynamic associated with
f(z)= z + z2
has infinitely many periodic points in any neighborhood of 0, hence, is not linearizable. 1942 1965 1988
19 / 55 20 / 55 KAM Theory Siegel’s Diophantine condition : Liouville numbers
Andrey Nikolaevich Kolmogorov J¨urgen Kurt Moser Siegel’s Diophantine condition on the rotation number ✓ is Vladimir Igorevich Arnold that a rational number p/q with a small denominator q cannot (1903 – 1987) (1928 – 1999) be too good of a rational approximation of ✓. (1937 – 2010)
The same condition was introduced by Liouville, who proved in 1844 that Siegel’s Diophantine condition is satisfied if ✓ is an algebraic number.
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Algebraic numbers Transcendental numbers A number which is not algebraic is transcendental. A complex number ↵ is algebraic if there exists a nonzero The existence of transcendental numbers was not known polynomial f Z[X] such that f(↵)=0. The smallest degree before 1844, when Liouville produced the first examples, like of such a polynomial2 is the degree of the algebraic number ↵. 1 ⇠ = 10n! · n 0 3 X For instance p2, i = p 1, p2, e2i⇡a/b (for a and b integers, The idea of Liouville is to prove a Diophantine property of b>0)arealgebraicnumbers. algebraic numbers, namely that rational numbers with small denominators do not produce sharp approximations. Hence, a The roots of the quintic polynomial real number with too good rational approximations cannot be algebraic. X5 6X +3 N! For instance, with the above number ⇠ and q =10 , are algebraic numbers (but cannot be expressed using radicals). N N! n! p 2 2 p = 10 , 0 < ⇠ < = q 10(N+1)! qN+1 · n=0 X 23 / 55 24 / 55 Liouville’s inequality (1844) The Diophantine condition of Liouville and Siegel
A real number ✓ satisfies a Diophantine condition if there Liouville’s inequality. Let ↵ Joseph Liouville exists a constant > 0 such that be an algebraic number of (1809 - 1882) p 1 degree d 2. There exists ✓ > q q c(↵) > 0 such that, for any p/q Q with q>0, 2 for all p/q Q with q 2. 2 p c(↵) ↵ > q qd A real number is a Liouville number if it does not satisfy a Diophantine condition.
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Generic vs full measure, Baire vs Lebesgue Mathematical genealogy Ren´eBaire Henri L´eonLebesgue (1874 – 1932) (1875 – 1941) Ren´eBaire (1899)
Arnaud Denjoy| (1909)
Charles Pisot| (1938)
Yvette Amice| (1965) In dynamical systems, a property is satisfied for a generic rotation number ✓ if it is true for all numbers in a countable Jean Fresnel| (1967) intersection of dense open sets – these sets are called G sets | by Baire who calls meager the complement of a G set. Michel Waldschmidt (1972) The set of numbers which do not satisfy a Diophantine condition is a generic set. For Lebesgue measure, the set of http://genealogy.math.ndsu.nodak.edu Liouville numbers (i.e. the set of numbers which do not satisfy aDiophantinecondition)hasmeasurezero. 27 / 55 28 / 55 Brjuno’s condition Improvements of Liouville’s inequality In terms of continued fraction, the Diophantine condition (of Liouville and Siegel)canbewritten In the lower bound log qn+1 p c(↵) sup < . ↵ > n 1 log qn 1 q qd The condition of Brjuno is for ↵ real algebraic number of degree d 3, the exponent d of q in the denominator of the right hand side was replaced by log q n+1 < . with q 1 n 1 n any > (d/2) + 1 by A. Thue (1909), X • 2pd by C.L. Siegel in 1921, • If a number ✓ satisfies the Diophantine condition, then it p2d by F.J. Dyson and A.O. Gel’fond in 1947, • any > 2 by K.F. Roth in 1955. satisfies Brjuno’s condition. But there are (transcendental) • numbers which do not satisfy the Diophantine condition, but satisfy Brjuno’s condition.
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Thue– Siegel– Roth Theorem Thue– Siegel– Roth Theorem
An equivalent statement is that, for any real algebraic Axel Thue Carl Ludwig Siegel Klaus Friedrich Roth irrational number ↵ and for any ✏ > 0, the set of p/q Q (1863 – 1922) (1896 – 1981) (1925 – 2015) such that 2 ✏ q q↵ p