Fractal uncertainty principle Semyon Dyatlov (MIT/Clay Mathematics Institute) joint work with Joshua Zahl (MIT) March 10, 2016 Semyon Dyatlov Fractal uncertainty principle March 10, 2016 1 / 14
Discrete uncertainty principle Discrete uncertainty principle We use the discrete case for simplicity of presentation Z N = Z / N Z = { 0 , . . . , N − 1 } ℓ 2 � u � 2 � j | u ( j ) | 2 N = { u : Z N → C } , N = ℓ 2 1 � k e − 2 π ijk / N u ( k ) F N u ( j ) = √ N The Fourier transform F N : ℓ 2 N → ℓ 2 N is a unitary operator Take X = X ( N ) , Y = Y ( N ) ⊂ Z N . Want a bound for some β > 0 N ≤ CN − β , � 1 X F N 1 Y � ℓ 2 N → ∞ (1) N → ℓ 2 Here 1 X , 1 Y : ℓ 2 N → ℓ 2 N are multiplication operators If (1) holds, say that X , Y satisfy uncertainty principle with exponent β Semyon Dyatlov Fractal uncertainty principle March 10, 2016 2 / 14
Discrete uncertainty principle Discrete uncertainty principle We use the discrete case for simplicity of presentation Z N = Z / N Z = { 0 , . . . , N − 1 } ℓ 2 � u � 2 � j | u ( j ) | 2 N = { u : Z N → C } , N = ℓ 2 1 � k e − 2 π ijk / N u ( k ) F N u ( j ) = √ N The Fourier transform F N : ℓ 2 N → ℓ 2 N is a unitary operator Take X = X ( N ) , Y = Y ( N ) ⊂ Z N . Want a bound for some β > 0 N ≤ CN − β , � 1 X F N 1 Y � ℓ 2 N → ∞ (1) N → ℓ 2 Here 1 X , 1 Y : ℓ 2 N → ℓ 2 N are multiplication operators If (1) holds, say that X , Y satisfy uncertainty principle with exponent β Semyon Dyatlov Fractal uncertainty principle March 10, 2016 2 / 14
Discrete uncertainty principle Basic properties N ≤ CN − β , � 1 X F N 1 Y � ℓ 2 N → ∞ ; β > 0 (2) N → ℓ 2 Why uncertainty principle? Semyon Dyatlov Fractal uncertainty principle March 10, 2016 3 / 14
Discrete uncertainty principle Basic properties � 1 X F N 1 Y F − 1 N ≤ CN − β , N � ℓ 2 N → ∞ ; β > 0 (2) N → ℓ 2 1 X localizes to X in position, F N 1 Y F − 1 localizes to Y in frequency N (2) = ⇒ these localizations are incompatible Semyon Dyatlov Fractal uncertainty principle March 10, 2016 3 / 14
Discrete uncertainty principle Basic properties N ≤ CN − β , � 1 X F N 1 Y � ℓ 2 N → ∞ ; β > 0 (2) N → ℓ 2 1 X localizes to X in position, F N 1 Y F − 1 localizes to Y in frequency N (2) = ⇒ these localizations are incompatible Volume bound using Hölder’s inequality: � 1 X F N 1 Y � ℓ 2 N ≤ � 1 X � ℓ ∞ N �F N � ℓ 1 N � 1 Y � ℓ 2 N → ℓ 2 N → ℓ 2 N → ℓ 1 N → ℓ ∞ N � | X | · | Y | ≤ N This norm is < 1 when | X | · | Y | < N . Cannot be improved in general: N = MK , X = M Z / N Z , Y = K Z / N Z = ⇒ � 1 X F N 1 Y � ℓ 2 N = 1 N → ℓ 2 Semyon Dyatlov Fractal uncertainty principle March 10, 2016 3 / 14
Application: spectral gaps Application: spectral gaps for hyperbolic surfaces ( M , g ) = Γ \ H 2 convex co-compact hyperbolic surface D 3 D 4 ℓ 3 / 2 ℓ 1 q 2 q 1 ℓ 2 F ℓ ℓ 1 / 2 M ℓ ℓ 2 / 2 γ 1 γ 2 q 3 ℓ 1 / 2 ℓ 2 / 2 ℓ 3 q 2 q 1 ℓ 3 / 2 D 1 D 2 Resonances: poles of the Selberg zeta function (with a few exceptions) ∞ s = 1 � � 1 − e − ( s + k ) ℓ � � Z M ( λ ) = , 2 − i λ ℓ ∈L M k = 0 where L M is the set of lengths of primitive closed geodesics on M Semyon Dyatlov Fractal uncertainty principle March 10, 2016 4 / 14
Application: spectral gaps Application: spectral gaps for hyperbolic surfaces ( M , g ) = Γ \ H 2 convex co-compact hyperbolic surface D 3 D 4 ℓ 3 / 2 ℓ 1 q 2 q 1 ℓ 2 F ℓ ℓ 1 / 2 M ℓ ℓ 2 / 2 γ 1 γ 2 q 3 ℓ 1 / 2 ℓ 2 / 2 ℓ 3 q 2 q 1 ℓ 3 / 2 D 1 D 2 Resonances: poles of the scattering resolvent � L 2 ( M ) → L 2 ( M ) , − ∆ g − 1 4 − λ 2 � − 1 Im λ > 0 � R ( λ ) = : L 2 comp ( M ) → L 2 loc ( M ) , Im λ ≤ 0 Existence of meromorphic continuation: Patterson ’75,’76, Perry ’87,’89, Mazzeo–Melrose ’87, Guillopé–Zworski ’95, Guillarmou ’05, Vasy ’13 Semyon Dyatlov Fractal uncertainty principle March 10, 2016 4 / 14
Application: spectral gaps Plots of resonances Three-funnel surface with ℓ 1 = ℓ 2 = ℓ 3 = 7 Data courtesy of David Borthwick and Tobias Weich See arXiv:1305.4850 and arXiv:1407.6134 for more Semyon Dyatlov Fractal uncertainty principle March 10, 2016 5 / 14
Application: spectral gaps Plots of resonances Three-funnel surface with ℓ 1 = 6, ℓ 2 = ℓ 3 = 7 Data courtesy of David Borthwick and Tobias Weich See arXiv:1305.4850 and arXiv:1407.6134 for more Semyon Dyatlov Fractal uncertainty principle March 10, 2016 5 / 14
Application: spectral gaps Plots of resonances Torus-funnel surface with ℓ 1 = ℓ 2 = 7, ϕ = π/ 2, trivial representation Data courtesy of David Borthwick and Tobias Weich See arXiv:1305.4850 and arXiv:1407.6134 for more Semyon Dyatlov Fractal uncertainty principle March 10, 2016 5 / 14
Application: spectral gaps The limit set and δ M = Γ \ H 2 hyperbolic surface Λ Γ ⊂ S 1 the limit set δ := dim H (Λ Γ ) ∈ ( 0 , 1 ) ℓ 1 ℓ 2 M ℓ ℓ 3 Trapped geodesics: those with endpoints in Λ Γ Semyon Dyatlov Fractal uncertainty principle March 10, 2016 6 / 14
Application: spectral gaps Spectral gaps Essential spectral gap of size β > 0: only finitely many resonances with Im λ > − β Application: exponential decay of waves (modulo finite dimensional space) Patterson–Sullivan theory: the topmost resonance is at λ = i ( δ − 1 2 ) , 0 , 1 � � where δ = dim H Λ Γ ∈ ( 0 , 1 ) ⇒ gap of size β = max 2 − δ δ > 1 δ < 1 2 2 Semyon Dyatlov Fractal uncertainty principle March 10, 2016 7 / 14
Application: spectral gaps Spectral gaps Essential spectral gap of size β > 0: only finitely many resonances with Im λ > − β Application: exponential decay of waves (modulo finite dimensional space) Patterson–Sullivan theory: the topmost resonance is at λ = i ( δ − 1 2 ) , 0 , 1 � � where δ = dim H Λ Γ ∈ ( 0 , 1 ) ⇒ gap of size β = max 2 − δ δ − 1 2 δ − 1 2 δ > 1 δ < 1 2 2 Semyon Dyatlov Fractal uncertainty principle March 10, 2016 7 / 14
Application: spectral gaps Spectral gaps Essential spectral gap of size β > 0: only finitely many resonances with Im λ > − β Application: exponential decay of waves (modulo finite dimensional space) Patterson–Sullivan theory: the topmost resonance is at λ = i ( δ − 1 2 ) , 0 , 1 � � where δ = dim H Λ Γ ∈ ( 0 , 1 ) ⇒ gap of size β = max 2 − δ Improved gap β = 1 2 − δ + ε for δ ≤ 1 / 2: Dolgopyat ’98, Naud ’04, Stoyanov ’11,’13, Petkov–Stoyanov ’10 Bourgain–Gamburd–Sarnak ’11, Oh–Winter ’14: gaps for the case of congruence quotients However, the size of ε is hard to determine from these arguments Semyon Dyatlov Fractal uncertainty principle March 10, 2016 7 / 14
Application: spectral gaps Spectral gaps via uncertainty principle Λ Γ ⊂ S 1 limit set, M = Γ \ H 2 , dim H Λ Γ = δ ∈ ( 0 , 1 ) Essential spectral gap of size β > 0: only finitely many resonances with Im λ > − β Theorem [D–Zahl ’15] Assume that Λ Γ satisfies hyperbolic uncertainty principle with exponent β . Then M has an essential spectral gap of size β − . Proof Enough to show e − β t decay of waves at frequency ∼ h − 1 , 0 < h ≪ 1 Microlocal analysis + hyperbolicity of geodesic flow ⇒ description of waves at times log ( 1 / h ) using stable/unstable Lagrangian states Hyperbolic UP ⇒ a superposition of trapped unstable states has norm O ( h β ) on trapped stable states Semyon Dyatlov Fractal uncertainty principle March 10, 2016 8 / 14
Application: spectral gaps Spectral gaps via uncertainty principle Λ Γ ⊂ S 1 limit set, M = Γ \ H 2 , dim H Λ Γ = δ ∈ ( 0 , 1 ) Essential spectral gap of size β > 0: only finitely many resonances with Im λ > − β Theorem [D–Zahl ’15] Assume that Λ Γ satisfies hyperbolic uncertainty principle with exponent β . Then M has an essential spectral gap of size β − . The Patterson–Sullivan gap β = 1 2 − δ corresponds to the volume bound: � | X | · | Y | | X | ∼ | Y | ∼ N δ ∼ N δ − 1 / 2 = ⇒ N Discrete UP with β for discretizations of Λ Γ ⇓ Hyperbolic UP with β/ 2 Semyon Dyatlov Fractal uncertainty principle March 10, 2016 8 / 14
Proving uncertainty principles Regularity of limit sets The sets X , Y coming from convex co-compact hyperbolic surfaces are δ -regular with some constant C > 0: C − 1 n δ ≤ � ≤ Cn δ , � � j ∈ X , 1 ≤ n ≤ N � X ∩ [ j − n , j + n ] Conjecture 1 If X , Y are δ -regular with constant C and δ < 1, then N ≤ CN − β , � 1 X F N 1 Y � ℓ 2 β = β ( δ, C ) > 0 N → ℓ 2 Implies that each convex co-compact M has essential spectral gap > 0 Conjecture holds for discrete Cantor sets with N = M k , k → ∞ � � � 0 ≤ ℓ< k a ℓ M ℓ � � a 0 , . . . , a k − 1 ∈ A X = Y = , A ⊂ { 0 , . . . , M − 1 } Semyon Dyatlov Fractal uncertainty principle March 10, 2016 9 / 14
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