Geodesic distances and intrinsic distances on some fractal sets Masanori Hino (Kyoto Univ.) International Conference on Advances on Fractals and Related Topics Chinese University of Hong Kong, December 11, 2012
1/17 1. Introduction M : a Riemaniann manifold d ( x , y ) : the intrinsic distance (or the Carnot– Carath´ eodory distance): f : Lipschitz on M , { } d ( x , y ) : = sup f ( y ) − f ( x ) . |∇ f | ≤ 1 a.e. This is equal to the geodesic distance ρ ( x , y ) : { the length of continuous curves } ρ ( x , y ) : = inf . connecting x and y
1/17 1. Introduction M : a Riemaniann manifold d ( x , y ) : the intrinsic distance (or the Carnot– Carath´ eodory distance): f : Lipschitz on M , { } d ( x , y ) : = sup f ( y ) − f ( x ) . |∇ f | ≤ 1 a.e. This is equal to the geodesic distance ρ ( x , y ) : { the length of continuous curves } ρ ( x , y ) : = inf . connecting x and y
1. Introduction （ cont’d ） 2/17 Intrinsic distance in the framework of Dirichlet forms (cf. Biloli–Mosco, Sturm etc.) ( K , λ ) : a locally compact, separable metric measure space ( E , F ) : a strong local regular Dirichlet form on L 2 ( K ; λ ) ◮ ( E , F ) is a closed, nonnegative-definite, symmetric bilinear form on L 2 ( K ; λ ) ; ◮ (Markov property) ∀ f ∈ F , ˆ f : = ( 0 ∨ f ) ∧ 1 ∈ F and E ( ˆ f , ˆ f ) ≤ E ( f , f ) ; ◮ (strong locality) For f , g ∈ F with compact support, if f is constant on a neighborhood of supp [ g ] , then E ( f , g ) = 0 .
1. Introduction （ cont’d ） 2/17 Intrinsic distance in the framework of Dirichlet forms (cf. Biloli–Mosco, Sturm etc.) ( K , λ ) : a locally compact, separable metric measure space ( E , F ) : a strong local regular Dirichlet form on L 2 ( K ; λ ) ◮ ( E , F ) is a closed, nonnegative-definite, symmetric bilinear form on L 2 ( K ; λ ) ; ◮ (Markov property) ∀ f ∈ F , ˆ f : = ( 0 ∨ f ) ∧ 1 ∈ F and E ( ˆ f , ˆ f ) ≤ E ( f , f ) ; ◮ (strong locality) For f , g ∈ F with compact support, if f is constant on a neighborhood of supp [ g ] , then E ( f , g ) = 0 .
1. Introduction （ cont’d ） 3/17 Intrinsic distance in the framework of Dirichlet forms (cf. Biloli–Mosco, Sturm etc.) ( K , λ ) : a locally compact, separable metric measure space ( E , F ) : a strong local regular Dirichlet form on L 2 ( K ; λ ) Typical example: R d , dx ) , ( K , λ ) = ( R R E ( f , g ) = 1 ∫ ∫ ∫ ∫ R d ( a ij ( x ) ∇ f ( x ) , ∇ g ( x )) R R d dx R 2 R R R d ) , for f , g ∈ F : = H 1 ( R R where ( a ij ( x )) d i , j = 1 is symmetric, uniformly positive and bounded.
1. Introduction （ cont’d ） 4/17 Intrinsic distance in the framework of Dirichlet forms (cf. Biloli–Mosco, Sturm etc.) ( K , λ ) : a locally compact, separable metric measure space ( E , F ) : a strong local regular Dirichlet form on L 2 ( K ; λ ) µ � f � : the energy measure of f ∈ F When f is bounded, ∫ ∫ ∫ ∫ K ϕ d µ � f � = 2 E ( f , f ϕ ) − E ( f 2 , ϕ ) ∀ ϕ ∈F ∩ C b ( K ) . If E ( f , g ) = 1 ∫ ∫ ∫ ∫ R d ( a ij ( x ) ∇ f ( x ) , ∇ g ( x )) R R d dx , then R 2 R R µ � f � ( dx ) = ( a ij ( x ) ∇ f ( x ) , ∇ f ( x )) R R d dx . R
1. Introduction （ cont’d ） 4/17 Intrinsic distance in the framework of Dirichlet forms (cf. Biloli–Mosco, Sturm etc.) ( K , λ ) : a locally compact, separable metric measure space ( E , F ) : a strong local regular Dirichlet form on L 2 ( K ; λ ) µ � f � : the energy measure of f ∈ F When f is bounded, ∫ ∫ ∫ ∫ K ϕ d µ � f � = 2 E ( f , f ϕ ) − E ( f 2 , ϕ ) ∀ ϕ ∈F ∩ C b ( K ) . If E ( f , g ) = 1 ∫ ∫ ∫ ∫ R d ( a ij ( x ) ∇ f ( x ) , ∇ g ( x )) R R d dx , then R 2 R R µ � f � ( dx ) = ( a ij ( x ) ∇ f ( x ) , ∇ f ( x )) R R d dx . R
1. Introduction （ cont’d ） 5/17 Intrinsic distance in the framework of Dirichlet forms (cf. Biloli–Mosco, Sturm etc.) ( K , λ ) : a locally compact, separable metric measure space ( E , F ) : a strong local regular Dirichlet form on L 2 ( K ; λ ) µ � f � : the energy measure of f ∈ F d ( x , y ) : the intrinsic distance f ∈ F loc ∩ C ( K ) { } d ( x , y ) : = sup f ( y ) − f ( x ) . and µ � f � ≤ λ In this framework, various Gaussian estimates of the transition density have been obtained.
1. Introduction （ cont’d ） 5/17 Intrinsic distance in the framework of Dirichlet forms (cf. Biloli–Mosco, Sturm etc.) ( K , λ ) : a locally compact, separable metric measure space ( E , F ) : a strong local regular Dirichlet form on L 2 ( K ; λ ) µ � f � : the energy measure of f ∈ F d ( x , y ) : the intrinsic distance f ∈ F loc ∩ C ( K ) { } d ( x , y ) : = sup f ( y ) − f ( x ) . and µ � f � ≤ λ In this framework, various Gaussian estimates of the transition density have been obtained.
1. Introduction （ cont’d ） 6/17 Questions: Is d identified with the geodesic distance (=shortest path metric)? In particular, what if K is a fractal set, which does not have a (usual) differential structure? But the straightforward formulation is not very useful as I will explain...
1. Introduction （ cont’d ） 6/17 Questions: Is d identified with the geodesic distance (=shortest path metric)? In particular, what if K is a fractal set, which does not have a (usual) differential structure? But the straightforward formulation is not very useful as I will explain...
7/17 2. Canonical Dirichlet forms on typical self- similar fractals Case of the 2-dim. standard Sierpinski gasket ⊃ K V 2 V n : n th level graph approximation ) n ( 5 E ( n ) ( f , f ) = ( f ( x ) − f ( y )) 2 ∑ ∑ ∑ 3 x , y ∈ V n , x ∼ y
7/17 2. Canonical Dirichlet forms on typical self- similar fractals Case of the 2-dim. standard Sierpinski gasket ⊃ K V 2 V n : n th level graph approximation n ( 5 ) E ( n ) ( f , f ) = ( f ( x ) − f ( y )) 2 ∑ ∑ ∑ 3 x , y ∈ V n , x ∼ y � � � scaling factor
2. Canonical Dirichlet forms on typical self-similar fractals （ cont’d ） 8/17 E ( n ) ( f , f ) ր ∃ E ( f , f ) ≤ + ∞ ∀ f ∈ C ( K ) . F : = { f ∈ C ( K ) | E ( f , f ) < + ∞ } Then, ( E , F ) is a strong local regular Dirichlet form on L 2 ( K ; λ ) . ( λ : the Hausdorff measure on K ) � � { X t } : “Brownian motion” on K � (invariant under scaling and isometric transformations) Similar construction is valid for more general finitely ramified fractals.
2. Canonical Dirichlet forms on typical self-similar fractals （ cont’d ） 8/17 E ( n ) ( f , f ) ր ∃ E ( f , f ) ≤ + ∞ ∀ f ∈ C ( K ) . F : = { f ∈ C ( K ) | E ( f , f ) < + ∞ } Then, ( E , F ) is a strong local regular Dirichlet form on L 2 ( K ; λ ) . ( λ : the Hausdorff measure on K ) � � { X t } : “Brownian motion” on K � (invariant under scaling and isometric transformations) Similar construction is valid for more general finitely ramified fractals.
2. Canonical Dirichlet forms on typical self-similar fractals （ cont’d ） 9/17 In many examples, µ � f � ⊥ λ (self-similar measure). Then, d ( x , y ) = sup { f ( y ) − f ( x ) | f ∈ F , µ � f � ≤ λ } = sup { f ( y ) − f ( x ) | f = const. } = 0. (This is closely connected with the fact that the heat kernel density has a sub-Gaussian estimate.) By taking different measures as λ , however, we have nontrivial quantities...
2. Canonical Dirichlet forms on typical self-similar fractals （ cont’d ） 9/17 In many examples, µ � f � ⊥ λ (self-similar measure). Then, d ( x , y ) = sup { f ( y ) − f ( x ) | f ∈ F , µ � f � ≤ λ } = sup { f ( y ) − f ( x ) | f = const. } = 0. (This is closely connected with the fact that the heat kernel density has a sub-Gaussian estimate.) By taking different measures as λ , however, we have nontrivial quantities...
2. Canonical Dirichlet forms on typical self-similar fractals （ cont’d ） 9/17 In many examples, µ � f � ⊥ λ (self-similar measure). Then, d ( x , y ) = sup { f ( y ) − f ( x ) | f ∈ F , µ � f � ≤ λ } = sup { f ( y ) − f ( x ) | f = const. } = 0. (This is closely connected with the fact that the heat kernel density has a sub-Gaussian estimate.) By taking different measures as λ , however, we have nontrivial quantities...
2. Canonical Dirichlet forms on typical self-similar fractals （ cont’d ） 10/17 K : 2-dim. Sierpinski gasket ( E , F ) : the standard Dirichlet form on L 2 ( K , ν ) with ν : = µ � h 1 � + µ � h 2 � (Kusuoka measure) ( h i : a harmonic function, E ( h i , h j ) = δ i , j ) Theorem (Kigami ’93, ’08, Kajino ’12) R 2 is homeomorphic; h : K → h h ( K ) ⊂ R ◮ (Ki) h h h R ◮ (Ka) The intrinsic distance d coincides with the geodesic h ( K ) by the identifying K and h h ( K ) ; h h distance ρ h h on h h ◮ (Ki, Ka) The transition density p ν t ( x , y ) has a h (= d ) ; Gaussian estimate w. r. t. ρ h h ◮ (Ki) The red line is the geodesic.
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