Multifractal analysis of arithmetic functions St´ ephane Jaffard Universit´ e Paris Est (France) Collaborators: Arnaud Durand Universit´ e Paris Sud Orsay Samuel Nicolay Universit´ e de Li` ege International Conference on Advances on Fractals and Related Topics Hong-Kong, December 10-14, 2012
Multifractal analysis Purpose of multifractal analysis : Introduce and study classification parameters for data (functions, measures, distributions, signals, images), which are based on regularity
Multifractal analysis Purpose of multifractal analysis : Introduce and study classification parameters for data (functions, measures, distributions, signals, images), which are based on regularity Fonction de Weierstrass : Espace x frequence = 1000x20 4 Weierstrass function 3 2 + ∞ 2 − Hj cos ( 2 j x ) � W H ( x ) = 1 0 j = 0 -1 0 < H < 1 -2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Multifractal analysis Purpose of multifractal analysis : Introduce and study classification parameters for data (functions, measures, distributions, signals, images), which are based on regularity Fonction de Weierstrass : Espace x frequence = 1000x20 4 Weierstrass function 3 2 + ∞ 2 − Hj cos ( 2 j x ) � W H ( x ) = 1 0 j = 0 -1 0 < H < 1 -2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Brownien : 1000 points 2.0 1.5 Brownian motion 1.0 0.5 0.0 -0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Everywhere irregular signals and images 70 Jet turbulence Eulerian velocity signal (ChavarriaBaudetCiliberto95) ∆ = 3.2 ms 35 0 0 300 temps (s) 600 900 Fully developed turbulence Internet Trafic 567 ! 689 !&( !&" !&' !&% !&$ !&# ! ! !&# ! !&$ ! "!! #!!! #"!! $!!! $"!! %!!! )*+,-./01234 Euro vs Dollar (2001-2009)
Van Gogh painting y f752 300 600 900 1200 1500 1800 2100 300 600 900 1200 1500
Van Gogh painting y y f752 50 100 300 150 200 250 600 300 350 400 450 900 500 100 200 300 400 500 1200 1500 1800 2100 300 600 900 1200 1500
Pointwise regularity Definition : Let f : R d → R be a locally bounded function and x 0 ∈ R d ; f ∈ C α ( x 0 ) if there exist C > 0 and a polynomial P such that, for | x − x 0 | small enough, | f ( x ) − P ( x − x 0 ) | ≤ C | x − x 0 | α
Pointwise regularity Definition : Let f : R d → R be a locally bounded function and x 0 ∈ R d ; f ∈ C α ( x 0 ) if there exist C > 0 and a polynomial P such that, for | x − x 0 | small enough, | f ( x ) − P ( x − x 0 ) | ≤ C | x − x 0 | α The H¨ older exponent of f at x 0 is f ∈ C α ( x 0 ) } h f ( x 0 ) = sup { α :
Pointwise regularity Definition : Let f : R d → R be a locally bounded function and x 0 ∈ R d ; f ∈ C α ( x 0 ) if there exist C > 0 and a polynomial P such that, for | x − x 0 | small enough, | f ( x ) − P ( x − x 0 ) | ≤ C | x − x 0 | α The H¨ older exponent of f at x 0 is f ∈ C α ( x 0 ) } h f ( x 0 ) = sup { α : The H¨ older exponent of the Weierstrass function W H is constant and equal to H (Hardy) The H¨ older exponent of Brownian motion is constant and equal to 1 / 2 (Wiener) W H and B are mono-H¨ older function
Multifractal spectrum (Parisi and Frisch, 1985) The iso-H¨ older sets of f are the sets E H = { x 0 : h f ( x 0 ) = H }
Multifractal spectrum (Parisi and Frisch, 1985) The iso-H¨ older sets of f are the sets E H = { x 0 : h f ( x 0 ) = H } Let f be a locally bounded function. The H¨ older spectrum of f is D f ( H ) = dim ( E H ) where dim stands for the Hausdorff dimension (by convention, dim ( ∅ ) = −∞ )
Multifractal spectrum (Parisi and Frisch, 1985) The iso-H¨ older sets of f are the sets E H = { x 0 : h f ( x 0 ) = H } Let f be a locally bounded function. The H¨ older spectrum of f is D f ( H ) = dim ( E H ) where dim stands for the Hausdorff dimension (by convention, dim ( ∅ ) = −∞ ) The upper-H¨ older sets of f are the sets E H = { x 0 : h f ( x 0 ) ≥ H } The lower-H¨ older sets of f are the sets E H = { x 0 : h f ( x 0 ) ≤ H }
Riemann’s non-differentiable function and beyond ∞ sin ( n 2 x ) � R 2 ( x ) = n 2 n = 1
Riemann’s non-differentiable function and beyond ∞ sin ( n 2 x ) � R 2 ( x ) = n 2 n = 1 4 H − 2 H ∈ [ 1 / 2 , 3 / 4 ] if d F ( H ) = 0 if H = 3 / 2 −∞ else
Riemann’s non-differentiable function and beyond ∞ sin ( n 2 x ) � R 2 ( x ) = n 2 n = 1 4 H − 2 H ∈ [ 1 / 2 , 3 / 4 ] if d F ( H ) = 0 if H = 3 / 2 −∞ else ∞ sin ( n 3 x ) � The cubic Riemann function : R 3 ( x ) = n 3 n = 1
Riemann’s non-differentiable function and beyond ∞ sin ( n 2 x ) � R 2 ( x ) = n 2 n = 1 4 H − 2 H ∈ [ 1 / 2 , 3 / 4 ] if d F ( H ) = 0 if H = 3 / 2 −∞ else ∞ sin ( n 3 x ) � The cubic Riemann function : R 3 ( x ) = n 3 n = 1 In a recent paper (arXiv :1208.6533v1) F . Chamizo and A. Ubis consider ∞ e iP ( n ) x � F ( x ) = deg ( P ) = k n α n = 1 Theorem : (Chamizo and Ubis) : let ν F be the maximal multiplicity of the zeros of P ′ . If 1 + k 2 < α < k and 1 k ( α − 1 ) ≤ H ≤ 1 α − 1 � � , k 2 then � � H − α − 1 d F ( H ) ≥ max ( ν f , 2 ) k
Generalization : Nonharmonic Fourier series Let ( λ n ) n ∈ N be a sequence of points in R d ; a nonharmonic Fourier series is a function f that can be written � a n e i λ n · x . f ( x ) =
Generalization : Nonharmonic Fourier series Let ( λ n ) n ∈ N be a sequence of points in R d ; a nonharmonic Fourier series is a function f that can be written � a n e i λ n · x . f ( x ) = The gap sequence associated with ( λ n ) is the sequence ( θ n ) : θ n = inf m � = n | λ n − λ m |
Generalization : Nonharmonic Fourier series Let ( λ n ) n ∈ N be a sequence of points in R d ; a nonharmonic Fourier series is a function f that can be written � a n e i λ n · x . f ( x ) = The gap sequence associated with ( λ n ) is the sequence ( θ n ) : θ n = inf m � = n | λ n − λ m | The sequence ( λ n ) is separated if : inf n θ n > 0 .
Generalization : Nonharmonic Fourier series Let ( λ n ) n ∈ N be a sequence of points in R d ; a nonharmonic Fourier series is a function f that can be written � a n e i λ n · x . f ( x ) = The gap sequence associated with ( λ n ) is the sequence ( θ n ) : θ n = inf m � = n | λ n − λ m | The sequence ( λ n ) is separated if : inf n θ n > 0 . Theorem : Let x 0 ∈ R d . If ( λ n ) is separated and f ∈ C α ( x 0 ) , then ∃ C such that ∀ n , C ( 1 ) if | λ n | ≥ θ n , then | a n | ≤ ( θ n ) α . Thus, if H = sup { α : ( 1 ) holds } , then, for any x 0 ∈ R d , h f ( x 0 ) ≤ H .
Generalization : Nonharmonic Fourier series Let ( λ n ) n ∈ N be a sequence of points in R d ; a nonharmonic Fourier series is a function f that can be written � a n e i λ n · x . f ( x ) = The gap sequence associated with ( λ n ) is the sequence ( θ n ) : θ n = inf m � = n | λ n − λ m | The sequence ( λ n ) is separated if : inf n θ n > 0 . Theorem : Let x 0 ∈ R d . If ( λ n ) is separated and f ∈ C α ( x 0 ) , then ∃ C such that ∀ n , C ( 1 ) if | λ n | ≥ θ n , then | a n | ≤ ( θ n ) α . Thus, if H = sup { α : ( 1 ) holds } , then, for any x 0 ∈ R d , h f ( x 0 ) ≤ H . Open problem : Optimality of this result
Davenport series The sawtooth function is � x − ⌊ x ⌋ − 1 / 2 if x �∈ Z { x } = 0 else ✻ 1 / 2 ✲ r r r 0 1 In one variable, Davenport series are of the form ∞ � a n { nx } , a n ∈ R . F ( x ) = n = 1
Spectrum estimates for Davenport series ∞ � a n { nx } , a n ∈ R . F ( x ) = n = 1 Assuming that ( a n ) ∈ l 1 , then F is continuous at irrational points and the jump at p / q (if p ∧ q = 1) is ∞ � B q = a nq n = 1
Spectrum estimates for Davenport series ∞ � a n { nx } , a n ∈ R . F ( x ) = n = 1 Assuming that ( a n ) ∈ l 1 , then F is continuous at irrational points and the jump at p / q (if p ∧ q = 1) is ∞ � B q = a nq n = 1 ∈ l ∞ and β > 1. Then Theorem : Assume that ( n β a n ) / dim ( E H ) ≥ H β if ( n β a n ) ∈ l ∞ and β > 2. Then dim ( E H ) ≤ 2 H β
Spectrum estimates for Davenport series ∞ � a n { nx } , a n ∈ R . F ( x ) = n = 1 Assuming that ( a n ) ∈ l 1 , then F is continuous at irrational points and the jump at p / q (if p ∧ q = 1) is ∞ � B q = a nq n = 1 ∈ l ∞ and β > 1. Then Theorem : Assume that ( n β a n ) / dim ( E H ) ≥ H β if ( n β a n ) ∈ l ∞ and β > 2. Then dim ( E H ) ≤ 2 H β Open problem : Sharpen these bounds
Hecke’s functions ∞ { nx } � H s ( x ) = n s . n = 1 The function H s ( x ) is a Dirichlet series in the variable s , and its analytic continuation depends on Diophantine approximation properties of x (Hecke, Hardy, Littlewood).
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