Heat content asymptotics of some domains with fractal boundary

Transcription

1 Heat content asymptotics of some domains with fractal boundary Philippe H. A. Charmoy Mathematical Institute, University of Oxford Partly based on joint work with D.A. Croydon and B.M. Hambly Cornell, June / 19

2 Outline 1 Motivation 2 Heat content estimates 3 Spatially homogenous snowflakes 4 Statistically self-similar snowflakes 5 Summary and further research 2 / 19

3 Motivation Motivation Let D R d, consider the solution u to the heat equation s u(s, x) = 1 u(s, x), 2 (s, x) (0, ) D, u(0, x) = u(0+, x) = 0, x D, u(, x) = 1, x D. The heat content is the total heat in D, i.e. E(s) = u(s, x)dx. D It is an alternative to the partition function to study how the geometry of D dictates its analytic behaviour. Brossard and Carmona, Can we hear the dimension of a fractal? See also van den Berg and Fleckinger et al. 3 / 19

4 Heat content estimates Inner Minkowski dimension Let D R d bounded and put µ(ɛ) = Leb({x D : dist(x, D) ɛ}). We can use this instead of the usual volume of the ɛ-sausages to define the inner Minkowski dimension of D dim M D = d lim sup ɛ 0 log µ(ɛ) log ɛ and dim M D = d lim inf ɛ 0 log µ(ɛ) log ɛ. When the dimension exists and is γ, we may define the inner Minkowski content as M = lim ɛ 0 ɛ γ d µ(ɛ), when it exists. 4 / 19

5 Heat content estimates Heat content estimates Intuition: solve the heat equation probabilistically using Brownian motion to get E(s) = P x (T D c s)dx. D Since B s s 1/2 for small s, we should expect E(s) to be of same order as 1 d(x, D) s 1/2dx = µ(s 1/2 ). D For example, if the inner Minkowski dimension of D exists and is equal to γ, then E(s) s (d γ)/2. More precisely... 5 / 19

6 Heat content estimates Theorem (van den Berg, 1994, Abelian ) Let D R 2 be simply connected. Then, c 1 µ(c 2 s 1/2 ) E(s) 2s 1 0 ɛe ɛ2 /4s µ(ɛ)dɛ for small s. In particular, if µ(ɛ) ɛ 2 γ, then E(s) s (2 γ)/2. Theorem (Upper and lower Minkowski dim.) Let D R 2 be simply connected. Then, ( ) d log E(s) lim inf = 1 s 0 2 log s 2 dim M D ( ) d log E(s) lim sup = 1 s 0 2 log s 2 dim M D. 6 / 19

7 Heat content estimates Self-affine curves We can use this to give another example disproving Berry s conjecture that the Hausdorff dimension should drive the analytic properties of the domain. The following self-affine curve has Hausdorff dimension and Minkowski dimension / 19

8 Spatially homogenous snowflakes Construction of spatially homogeneous snowflakes Construction of spatially homogeneous snowflakes Let A N bounded. For a A, we consider the curve made of m(a) = 3a + 1 segments of length l(a) 1 = (2a + 1) 1 arranged to produce a spikes, e.g. K(2) and K(3) are as shown. For a sequence ξ = (ξ n ) in A, we use these building blocks to create a Koch curve K(ξ) iteratively. 8 / 19

9 Spatially homogenous snowflakes Construction of spatially homogeneous snowflakes For example, K(1, 3, 2, 1,... ) looks as follows. At step n, we have M n = n i=1 m(ξ i ) linear pieces of size L 1 n = n l(ξ i ) 1. i=1 We are interested in the simply connected domain D = D(ξ) enclosed by three copies of K(ξ). 9 / 19

10 Spatially homogenous snowflakes Geometry Geometry By counting the area under the spikes at level n, we get µ(l 1 n ) M n L 2 n. Sure enough, we can find sequences such that the inner Minkowski dimension of D does not exist. But when (ξ n ) is i.i.d. then dim M D = lim n thanks to the strong law of large numbers. log M n log L n, a.s. 10 / 19

11 Spatially homogenous snowflakes Geometry Finer fluctuations are recovered using the LIL... Theorem Suppose that (ξ n ) is i.i.d. Then, c 1 ɛ 2 γ e c 2ψ(1/ɛ) µ(ɛ) c 3 ɛ 2 γ e c 2ψ(1/ɛ) for small ɛ, where γ = dim M D and ψ(x) = log x log log log x. Furthermore, lim inf ɛ 0 µ(ɛ)e c 4ψ(1/ɛ) ɛ 2 γ < and lim sup ɛ 0 µ(ɛ)e c 4ψ(1/ɛ) ɛ 2 γ > / 19

12 Spatially homogenous snowflakes Heat content Heat content We can feel the upper and lower Minkowski dimensions. When (ξ n ) is i.i.d., we can feel the LIL. Theorem Suppose that (ξ n ) is i.i.d. Then, c 1 s 1 γ/2 e c 2ψ(1/s) E(s) c 3 s 1 γ/2 e c 2ψ(1/s) for small s, where γ = dim M D and ψ(x) = log x log log log x. Furthermore, lim inf s 0 E(s)e c 4ψ(1/s) s 1 γ/2 < and lim sup s 0 E(s)e c 4ψ(1/s) s 1 γ/2 > / 19

13 Statistically self-similar snowflakes Statistically self-similar snowflakes We can add spatial randomness in the construction above... The fractal is encoded by the general branching process. We think of sets of size e t as offspring born at time t. Its Minkowski dimension γ, say, exists and can be calculated using suitable branching process techniques. U 1 V 13 / 19

14 Statistically self-similar snowflakes Heat content Heat content Consider a third of the snowflake where we keep the linear part of the boundary at temperature 0 and write F U (s) for the total heat in it at time s. By scaling, F U (s) = i F Ri U i (s) + η(s) = i Ri 2 F Ui (R 2 i s) + η(s), where η is a small error due to keeping the polygonal region at temperature 0. U 1 V 14 / 19

15 Statistically self-similar snowflakes Heat content Using the principle of not feeling the boundary and renewal theory... Theorem We have s 1 γ/2 E(s) m 1 W, a.s. for some positive m 1 and random variable W with EW = 1 which encodes the macro randomness. In fact, there is a similar result for µ. In particular the inner Minkowski dimension is γ and the inner Minkowski content exists. Further, both can be recovered from E. 15 / 19

16 Summary and further research Summary Summary We can recover the upper and lower Minkowski dimensions of the boundary from E, and in particular know whether they are equal. For the spatially homogeneous snowflakes generated with an i.i.d. sequence, log µ fluctuates according to the LIL and there is no Minkowski content. The same fluctuations occur for E For the statistically self-similar snowflakes, both the Minkowski dimensions and content exist, and both can be recovered from E. 16 / 19

17 Summary and further research Open question Open question Given the spatial independence of the boundary of the statistically self-similar snowflake, are the fluctuations around the limiting behaviour given by a CLT? For some open subsets of [0, 1] we can. 17 / 19

18 Summary and further research Open question This suggests that s γ/4 [s γ/2 1 E(s) m 1 W ] d Z, where E [ ] [ ] e iθz = E e 1 2 θ2 σ 2 W for some σ (0, ). Intuitively, this would imply that E(s) m 1 s 1 γ/2 W + m 2 s 1 γ/4 Z + o(s 1 γ/4 ). Compare with the deterministic triadic snowflake E(s) p(s)s 1 γ/2 + q(s)s + o(s), for some log 9-periodic functions p and q. 18 / 19

19 Summary and further research Open question Idea of the proof Iterating F U (s) = i F Ri U i (s) + η(s) = i Ri 2 F Ui (R 2 i s) + η(s), yields F U (s) = x C Rx 2 F Ux (Rx 2 s) + η C (s), where C is a cut of the trace of the branching process such that the sets R x U x have roughly the same size. For an appropriate cut, the accumulated error η C can be controlled. Then a Taylor expansion argument shows that the sum converges in distribution to the desired limit. 19 / 19

20 Summary and further research Open question Some references Brossard and Carmona 1986, Can one hear the dimension of a fractal. Charmoy, preprint, Heat content asymptotics of some Koch type snowflakes. Charmoy, Croydon and Hambly, in preparation, Central limit theorems for the spectra of random self-similar fractals with Dirichlet weights. Kac 1966, Can one hear the shape of a drum? van den Berg 1994, Heat content and Brownian motion for some regions with a fractal boundary. THANK YOU 20 / 19