Main results Other examples On a recursive construction of Dirichlet form on the Sierpi´ nski gasket Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) The Chinese University of Hong Kong qsgu@math.cuhk.edu.hk June 2017 Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Background Recall that a Sierpi´ nski gasket is the unique non-empty compact set K in R 2 satisfying K = � 3 i =1 F i ( K ) for an iterated function i =1 on R 2 such that F i ( x ) = 1 system (IFS) { F i } 3 2 ( x − p i ) + p i with π √− 1 � � p 1 = 0, p 2 = 1, p 3 = exp . 3 Denote by V 0 = { p 1 , p 2 , p 3 } the boundary of K , and let F ω = F ω 1 ◦ · · · ◦ F ω n for a word ω ∈ W n = { 1 , 2 , 3 } n . Let ω ∈ W n F ω ( V 0 ) and V ∗ = � ∞ V n = � n =0 V n . Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Background K = V ∗ under the Euclidean metric. 1 The Hausdorff dimension of K is α = log 3 log 2 . 2 The standard Dirichlet form ( E , F ) on L 2 ( K .ν ) is 3 well-known: � n � � 5 ( u ( p ) − u ( q )) 2 , E ( u ) = lim (1) 3 n →∞ p ∼ n q F = { u ∈ C ( K ) : E ( u ) < ∞} (2) where p ∼ n q means p � = q and p , q ∈ F ω ( V 0 ) for some word ω ∈ W n . Self-similar identity : 4 3 1 � E ( u ) = E ( u ◦ F i ) , (3) r i i =1 where r i > 0, i = 1 , 2 , 3 are called renormalization factors . Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Background How about Dirichlet forms without self-similar identity? For n ≥ 0, let Γ n be the graph on V n with edge relation ∼ n , for u ∈ ℓ ( V n ), let ( E n , ℓ ( V n )) be c ( n ) � pq ( u ( p ) − u ( q )) 2 , E n ( u ) = p ∼ n q where c ( n ) pq ≥ 0 are conductances . Compatible : the restriction of E n on ℓ ( V n − 1 ) must be E n − 1 . In [Meyers,Strichartz,Teplyaev 2004], the authors use the compatible condition to solve the equations of conductances from a given harmonic function. Now let us consider a special case, that is we require the conductances of the cells F ω ( V 0 ) on the same level | ω | = n are the same. We call this kind of construction the recursive construction . Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Background Let ( a 0 , b 0 , c 0 ) be the conductance on V 0 , and let ( a n , b n , c n ) be the conductances of F ω ( V 0 ) , | ω | = n , n ≥ 1 to be determined. ∆ − Y transform gives ( x n , y n , z n ) on the Y -side. Compatibility: x n − 1 = x n + φ ( x n ; y n , z n ) , y n − 1 = y n + φ ( y n ; z n , x n ) , n ≥ 1 , (4) z n − 1 = z n + φ ( z n ; x n , y n ) , where φ ( x n ; y n , z n ) := ( x n + y n )( x n + z n ) 2( x n + y n + z n ) . Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Construction Proposition For a 0 , b 0 , c 0 > 0 , in order that (4) to have positive solution ( x n , y n , z n ) , n ≥ 1 , it is necessary and sufficient that x 0 ≥ y 0 = z 0 > 0 (or the symmetric alternates). In this case, x n ≥ y n = z n > 0 , n ≥ 0 and { ( x n , y n , z n ) } n ≥ 0 is uniquely determined by the initial data ( x 0 , y 0 , z 0 ) . Figure: Consistence of the n -th and ( n − 1)-th resistance networks Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Construction Sketch of the Proof. Sufficiency . � � � 1 4 x 2 0 + 6 x 0 y 0 + 6 y 2 x 1 = 14 x 0 + 3 y 0 − 2 , 15 0 (5) � � � y 1 (= z 1 ) = 1 4 x 2 0 + 6 x 0 y 0 + 6 y 2 − 2 x 0 + y 0 + , 0 5 Necessity . Assume that x 0 ≥ y 0 > z 0 . We show that x n ≫ y n ≫ z n and from this we deduce that z n < 0, a contradiction. Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Construction Recall that the (effective) resistance R := R ( a 0 , b 0 ) on V ∗ × V ∗ is defined for any two distinct points x , y ∈ V ∗ , R ( x , y ) − 1 := inf {E ( u ) : u ∈ ℓ ( V ∗ ) , u ( x ) = 1 , u ( y ) = 0 } . Proposition For a 0 > b 0 = c 0 , the completion of the ( V ∗ , R ( a 0 , b 0 ) ) is K, and C − 1 | x − y | ≤ R ( a 0 , b 0 ) ( x , y ) ≤ C | x − y | γ ′ , x , y ∈ K where γ ′ = log 3 log 2 − 1 and C > 0 is a constant depends on a 0 and b 0 . Furthermore R ( a 0 , b 0 ) is a bounded metric with � � R ( a 0 , b 0 ) ( x , y ) : x , y ∈ K ≤ C ′ b − 1 sup 0 . (6) where C ′ > 0 is independent of a 0 and b 0 . Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Construction Theorem For the case x 0 > y 0 = z 0 > 0 in the above theorem, we have � n � 3 x n 1 y n (2 x n + y n ) ≍ 2 n , a n = b n = c n = ≍ . 2 x n + y n 2 n →∞ E ( a 0 , b 0 ) Moreover E ( a 0 , b 0 ) ( u ) = lim ( u ) defines a strongly local n regular Dirichlet form on L 2 ( K , µ ) ; it satisfies 3 E ( a 0 , b 0 ) ( u ) = � E ( a 1 , b 1 ) ( u ◦ F i ) (7) i =1 but does NOT satisfy the energy self-similar identity. Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Construction The dichotomic situation: case 1: x 0 = y 0 . the standard Dirichlet form[J.Kigami, 1 Analysis on fractals]. case 2: x 0 > y 0 . First created by 2 [K.Hattori,T.Hattori,H.Watanabe1994], they call it the asymptotically one-dimensional diffusion processes on the SG (later studied by [B.Hambly,T.Kumagai1996], [B.Hambly,O.Jones 2002], [B.Hambly,W.Yang arXiv1612.02342]). Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Spectrum asymptotic For the Dirichlet form ( E ( a 0 , b 0 ) , F ) with a 0 > b 0 , we study the distribution of the eigenvalues. Let ∆ ( a 0 , b 0 ) be the Laplacian , the E ( a 0 , b 0 ) , F on L 2 ( K , µ ), where µ is � � infinitesimal generator of fixed to be the normalized α -Hausdorff measure. Denote by ρ ( a 0 , b 0 ) ( t ) the eigenvalue count of the Dirichlet boundary condition (D.B.C), that is � λ ≤ t : λ is an eigenvalue of − ∆ ( a 0 , b 0 ) with D.B.C. � ρ ( a 0 , b 0 ) ( t ) = # , (8) and denote by ρ ( a 0 , b 0 ) ( t ) the count with Neumann boundary N condition (N.B.C). Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Spectrum asymptotic Our main result is the following. Theorem Assume that a 0 > b 0 = c 0 , then log 3 log(9 / 2) , ρ ( a 0 , b 0 ) ( t ) ≍ t t → ∞ . This estimate improves the lower bound of ρ ( a 0 , b 0 ) ( t ) in [B.Hambly,O.Jones2002,Theorem 13] where it was shown to be C − 1 t log 3 / log(9 / 2) (log t ) − β with β > log 3 / log 2, using a heat kernel technique in the estimation. Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Spectrum asymptotic The first basic lemma:( bounds of the λ 1 ) Lemma There exists C > 0 such that for any initial data a > b = c > 0 on Γ 0 , we have C − 1 b ≤ λ ( a , b ) ≤ Cb , (9) 1 where λ ( a , b ) is the first eigenvalues of − ∆ ( a , b ) with the Dirichlet 1 boundary condition. Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Spectrum asymptotic Sketch of proof We will make use of the Rayleigh quotient for the first eigenvalue: E ( u ) λ 1 = inf , (10) || u || 2 u ∈F 0 , u � =0 2 where F 0 := { u ∈ F : u | V 0 = 0 } . Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Spectrum asymptotic The 1-harmonic function u 1 gives the upper bound of λ 1 , Figure: the value of u 1 and the uniform upper bound of R ( a , b ) gives the lower bound of λ 1 . Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
Main results Other examples Spectrum asymptotic The Second lemma:( scaling property) Lemma Let a 0 > b 0 = c 0 , then for all t ≥ 0 and n ≥ 0 , 3 n ρ ( a n , b n ) � t � t � ρ ( a 0 , b 0 ) ( t ) ≤ 3 n ρ ( a n , b n ) � ≤ ρ ( a 0 , b 0 ) ( t ) , and . N N 3 n 3 n (11) For the idea, we refer to the similar proof in [J.Kigami,M.Lapidus1993,Propositions 6.2, 6.3], in which, they use the self-similar identity (3). Here we use the identity (7). Qingsong Gu (Joint work with Ka-Sing Lau and Hua Qiu) On a recursive construction of Dirichlet form on the Sierpi´ nski ga
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