Multifractal analysis: an example with two different Olsens cutoff - PowerPoint PPT Presentation

Multifractal analysis: an example with two different Olsen’s cutoff functions Jacques Peyri` ere, Paris-Sud University and BUAA CUHK, December 14, 2012 Setting 1 General results 2 An example 3 Joint work with Fathi Ben Nasr to appear in Revista Matem´ atica Iberoamericana Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 1 / 28

Besicovitch spaces ( X , d): a metric space having the Besicovitch property: There exists an integer constant C B such that one can extract C B countable � � { B j , k } k families 1 ≤ j ≤ C B from any collection B of balls so that � B j , k contains the centers of the elements of B , 1 j , k for any j and k � = k ′ , B j , k ∩ B j , k ′ = ∅ . 2 B( x , r ) stands for the open ball B( x , r ) = { y ∈ X ; d( x , y ) < r } . The letter B with or without subscript will implicitly stand for such a ball. When dealing with a collection of balls { B i } i ∈ I the following notation will implicitly be assumed: B i = B( x i , r i ). Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 2 / 28

Coverings and packings δ -cover of E ⊂ X : a collection of balls of radii not exceeding δ whose union contains E . A centered cover of E is a cover of E consisting in balls whose centers belong to E . δ -packing of E ⊂ X : a collection of disjoint balls of radii not exceeding δ centered in E . Besicovitch δ -cover of E ⊂ X : a centered δ -cover of E which can be decomposed into C B packings. Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 3 / 28

Packing measures and dimension �� � t r t δ ( E ) = sup j ; { B j } δ -packing of E , P t ( E ) t = lim δ ( E ) , P δ ց 0 P �� � � t ( E j ) ; E ⊂ P t ( E ) = inf , P E j t ( E ) = 0 } = sup { t ∈ R ; P t ( E ) = ∞} ∆( E ) = inf { t ∈ R ; P inf { t ∈ R ; P t ( E ) = 0 } = sup { t ∈ R ; P t ( E ) = ∞} dim P E = One has ∆( E ) = dim B E . Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 4 / 28

Centered Hausdorff measures �� � t r t H δ ( E ) = inf j ; { B j } centered δ -cover of E , t ( E ) t H = δ ց 0 H lim δ ( E ) , � � t ( F ) ; F ⊂ E H t ( E ) = sup . H dim H E = inf { t ∈ R ; H t ( E ) = 0 } = sup { t ∈ R ; H t ( E ) = ∞} Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 5 / 28

Lower bounds for dimensions ν : a non-negative function defined on the set of balls of X . �� � ν δ ( E ) = inf ν (B j ) : { B j } centered δ -cover of E ν ( E ) = δ ց 0 ν δ ( E ) lim ν ♯ ( E ) = sup ν ( F ) F ⊂ E Lemma If ν ♯ ( E ) > 0 , then � � log ν B( x , r ) dim H E ≥ x ∈ E , ν ♯ lim inf , (1) ess sup log r r ց 0 � � log ν B( x , r ) dim P E ≥ x ∈ E , ν ♯ lim sup , (2) ess sup log r r ց 0 Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 6 / 28

� � log ν B( x , r ) To prove (1), take γ < ess sup x ∈ E , ν ♯ lim inf r ց 0 and consider the set log r � � � � log ν B( x , r ) . We have ν ♯ ( F ) > 0. For all x ∈ F , F = x ∈ E ; lim inf r ց 0 > γ log r � � ≤ r γ . Consider the there exists δ > 0 such that, for all r ≤ δ , one has ν B( x , r ) set � � � ≤ r γ � F ( n ) = x ∈ F ; ∀ r ≤ 1 / n , ν B( x , r ) . We have F = � n ≥ 1 F ( n ). Since ν ♯ ( F ) > 0, there exists n such that ν ♯ � � F ( n ) > 0, and therefore there is a subset G of F ( n ) such that ν ( G ) > 0. Then for any centered δ -cover { B j } of G , with δ ≤ 1 / n , one has � � r γ ν δ ( G ) ≤ ν (B j ) ≤ j . Therefore, γ ν δ ( G ) ≤ δ ( G ) , H and γ ( G ) ≤ H γ ( G ) , 0 < ν ( G ) ≤ H which implies dim H E ≥ dim H G ≥ γ . Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 7 / 28

� � log ν B( x , r ) To prove (2), take γ < ess sup x ∈ E , ν ♯ lim sup r ց 0 and consider the set log r � � � � log ν B( x , r ) . We have ν ♯ ( F ) > 0, so there exists F = x ∈ E ; lim sup r ց 0 > γ log r a subset F ′ of F such that ν ( F ′ ) > 0. Let G be a subset of F ′ . Then, for � � ≤ r γ . Then for all x ∈ G , for all δ > 0, there exists r ≤ δ such that ν B( x , r ) � � all δ , by using the Besicovitch property, there exists a collection { B j , k } j 1 ≤ k ≤ C B of δ -packings of G which together cover G and such that ν (B j , k ) ≤ r γ j , k . Then one has � � r γ ν δ ( G ) ≤ ν (B j , k ) ≤ j , k . j , k � j , k ≥ 1 r γ This implies that there exists k such that ν δ ( G ). So we have C B j C B ν ( G ). So if F ′ = � G j , one has γ γ ( G ) ≥ 1 1 δ ( G ) ≥ C B ν δ ( G ). This implies P P � � γ ( G j ) ≥ 1 ν ( G j ) ≥ 1 ν ( F ′ ) > 0 , P C B C B so P γ ( F ′ ) > 0. Therefore, dim P F ≥ γ . Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 8 / 28

Level sets of local H¨ older exponents µ : a non-negative function of balls of X such that µ (B) = 0 and B ′ ⊂ B = ⇒ µ (B ′ ) = 0. S µ , the support of µ , is the complement of � µ (B)=0 B. � � � � log µ B( x , r ) x ∈ S µ ; lim sup ≤ α X µ ( α ) = , log r r ց 0 � � � � log µ B( x , r ) X µ ( α ) = x ∈ S µ ; lim inf ≥ α , log r r ց 0 X µ ( α, β ) = X µ ( α ) ∩ X µ ( β ) , and X µ ( α ) = X µ ( α ) ∩ X µ ( α ) . Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 9 / 28

Olsen’s packing measures � � ∗ � q , t j µ (B j ) q ; { B j } δ -packing of E r t P µ,δ ( E ) = sup , where ∗ means that one only sums the terms for which µ (B j ) � = 0, q , t q , t P µ ( E ) = δ ց 0 P lim µ,δ ( E ) , �� � � q , t P q , t µ ( E ) = inf P µ ( E j ) ; E ⊂ , E j q , t q , t τ µ ( q ) = inf { t ∈ R ; P µ (S µ ) = 0 } = sup { t ∈ R ; P µ (S µ ) = ∞} inf { t ∈ R ; P q , t µ (S µ ) = 0 } = sup { t ∈ R ; P q , t B µ ( q ) = µ (S µ ) = ∞} τ µ and B µ are convex. Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 10 / 28

Alternate definition of τ µ Fix λ < 1 and define � � � ∗ � m � µ k (B j ) q k ; { B j } packing of E with λδ < r j ≤ δ P q , t r t µ,δ ( E ) = sup , j k =1 � � P q , t P q , t µ ( E ) = lim µ,δ ( E ) , δ ց 0 and � � t ∈ R ; � P q , t µ ( E ) = + ∞ τ µ, E ( q ) � = sup . Proposition For any λ < 1 , one has � τ µ, S µ = τ µ and τ µ ( q ) = � � � m ∗ � − 1 µ k (B j ) q k ; { B j } packing of S µ with λδ < r j ≤ δ lim log δ log sup . δ ց 0 k =1 Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 11 / 28

Olsen’s Hausdorff measures � � � ∗ q , t j µ (B j ) q ; { B j } centered δ -cover of E r t H µ,δ ( E ) = inf , q , t q , t H µ ( E ) = δ ց 0 H lim µ,δ ( E ) , � � q , t H q , t ( E ) = sup H µ ( F ) ; F ⊂ E . µ b µ ( q ) = inf { t ∈ R ; H q , t (S µ ) = 0 } = sup { t ∈ R ; H q , t (S µ ) = ∞} µ µ In general, b µ is not convex. One always has b µ ≤ B µ ≤ τ µ . Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 12 / 28

Legendre transform: f ∗ ( y ) = inf x ∈ R xy + f ( x ). Theorem (Olsen, Ben Nasr-Bhouri-Heurteaux) dim H X α ≤ b ∗ ( α ) . 1 dim P X α ≤ B ∗ ( α ) . 2 If − α = B ′ ( q ) exists and dim H X α = B ∗ ( q ) , then B ( q ) = b ( q ) . 3 If for some q, H q , B ( q ) (S µ ) > 0 and − α = B ′ ( q ) exists, then 4 µ r ∈ R B ( r ) + α r = B ( q ) − qB ′ ( q ) . dim H X ( α ) = inf Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 13 / 28

Main lemma � � � ∗ q , t r t j µ (B j ) q ν (B j ) ; { B j } δ -packing of E Q µ,ν,δ ( E ) = sup , q , t q , t Q µ,ν ( E ) = δ ց 0 Q lim µ,ν,δ ( E ) , �� � � Q µ,ν ( E ) = inf Q µ,ν ( E j ) : E ⊂ . E j q , t q , t ϕ µ,ν ( q ) = inf { t ∈ R ; Q µ,ν (S µ ) = 0 } = sup { t ∈ R ; Q µ,ν (S µ ) = ∞} inf { t ∈ R ; Q q , t µ,ν (S µ ) = 0 } = sup { t ∈ R ; Q q , t ϕ µ,ν ( q ) = µ,ν (S µ ) = ∞} Lemma Assume that ϕ µ,ν (0) = 0 and ν ♯ (S µ ) > 0 . Then one has ν ♯ � �� � C X µ − ϕ ′ r (0) , − ϕ ′ l (0) = 0 , The same result holds with ϕ µ,ν . Jacques Peyri` ere (F. Ben Nasr & J. Peyri` ere) Multifractal b � = B CUHK, December 14, 2012 14 / 28