Nature is an infinite sphere of which the center is everywhere and the circumference nowhere. Blaise Pascal

Transcription

1 Nature is an infinite sphere of which the center is everywhere and the circumference nowhere. Blaise Pascal

2 p-adic Cantor strings and complex fractal dimensions Michel Lapidus, and Machiel van Frankenhuijsen Department of Mathematics Hawai i Pacific University 6Cornell Conference on Analysis, Probability and Mathematical Physics on Fractals Ithaca NY

3 Cantor set Henry John Stephen Smith discovered the Cantor set in Georg Ferdinand Ludwig Philipp Cantor introduced the Cantor set as an example of a perfect set that is nowhere dense in the real line R in Ternary Cantor set C = c 2 [0, 1] :c = a 0 + a 1 3 +a , a i 2{0, 2}for all i 0 C is self-similar: C = ' 1 (C) [ ' 2 (C) where ' 1 (x) = x 3 and ' 1(x) = x

4 Cantor fractal string Michel Lapidus considered the complement of the Cantor set in [0,1] as an infinite sequence of lengths in The ordinary Cantor string CS = 1 3, 1 9, 1 27,...with the corresponding multiplicity 1, 2, 4,... Let s 2 C and consider the geometric zeta function associated with the Cantor string CS (s) = 1 3 s s s + = 1 3 s 1X n=0 2 n 1 = 3 s 3 s 2 Complex dimensions are poles of the geometric zeta function. They are! = log 2 2 log 3 + in log 3 = D + inp, n 2 Z, for the Cantor string CS.

5 Complex fractal dimensions CS = log 2 log 3 = D M where D M = inf{ 0 : V CS (") =O(" 1 ) as "! 0 + } is the Minkowski dimension of the Cantor string and CS = inf{ 2 R : P 1 n=1 m n l n < 1} is the abscissa of convergence of the Dirichlet series defining the geometric zeta function CS. Theorem (M. L. Lapidus) Let L be a real fractal string with infinitely many nonzero lengths, then L = D M. Complex dimensions reveal oscillations intrinsic to the geometry, spectrum and dynamic of the fractal string.

6 Kurt Hensel field of p-adic numbers Q p is the completion of Q wrt the p-adic norm p : Q! [0, 1) given by x p = p v and 0 p = 0. (Q p, p ) is an ultrametric space since x + y p apple max{ x p, y p }. (Q p, p ) is a nonarchimedean field since x + x p apple x p. (Q p, p ) is locally compact and totally disconnected. (R, )=(Q 1, 1 ) is the archimedean field at infinity. The topological boundary of a p-adic ball is empty and every point in the p-adic ball is a center!

7 Alexander Ostrowski Theorem (Ostrowski Theorem) Every completion of Q is equivalent to Q p for some prime p apple1. Q p = {a v p v + + a 0 + a 1 p + a 2 p 2 + v 2 Z, a i 2 {0, 1, 2, 3,...,p 1}} The unit ball in Q p is the ring of p-adic integers Z p = {a 0 + a 1 p + a 2 p 2 + a i 2{0, 1, 2, 3,...,p 1}}

8 Nonarchimedean 3-adic Cantor set and string The 3-adic Cantor set C 3 is the self-similar set generated by the family of similarity contraction mappings { 1 (x) =3x, 2(x) =3x + 2} of Z 3 into itself. C 3 = {x 2 Z 3 x = a 0 + a a , a i 2 {0, 2} for all i 0} C 3 is naturally homeomorphic to the ternary Cantor set C. The 3-adic Cantor string CS 3 is the complement of C 3 in Z 3. CS 3 =(1 + 3Z 3 ) [ (3 + 9Z 3 ) [ (5 + 9Z 3 ) [ is isometric to the archimedean Cantor string CS. Complex dimensions of CS 3 are! = log 2 2 log 3 + in log 3

9 5-adic Cantor set and string by Chugh, Kumar & Rani The 5-adic Cantor set C 5 is the self-similar set generated by the family of similarity contraction mappings { 1 (x) =5x, 2(x) =5x + 2, 3(x) =5x + 4}. C 5 = {x 2 Z 5 x = a 0 + a a , a i 2 {0, 2, 4} for all i 0} The nonarchimedean 5-adic Cantor set C 5 is homeomorphic to the archimedean quinary Cantor set C 5. CS 5 =(1 + 5Z 5 ) [ (3 + 5Z 5 ) [ (5 + 25Z 5 ) [ ( Z 5 ) [ is isometric to the archimedean quinary Cantor string CS 5. Complex dimensions of the 5-adic Cantor string CS 5 are! = log 3 2 log 5 + in log 5, n 2 Z.

10 p-adic Cantor sets and strings For p > 2, the p-adic Cantor set C p is the self-similar set generated by the family of similarity contraction mappings { 1 (x) =px, 2(x) =px + 2,..., p+1 (x) =px + p 1} 2 C p = {x 2 Z p x = a 0 + a 1 p + a 2 p 2 +, a i 2 {0, 2,...,p 1} for all i 0} The nonarchimedean p-adic Cantor set C p is homeomorphic to the archimedean pinary Cantor set C p The p-adic Cantor string CS p is the complement of the p-adic Cantor set C p in Z p The geometric zeta function of the p-adic Cantor string is CSp (s) = p 1 2p s p 1 Complex dimensions of CS p are! = the residue of CSp at! is p 1 (p+1) log p log p+1 2 log p 2 + in log p and

11 Complex fractal dimensions Theorem Let L p be a p-adic fractal strings with infinitely many nonzero lengths, then Lp = D M. Complex dimensions reveal oscillations in the geometry of p-adic fractal strings

12 Exact tube formula for self-similar strings V CSp (") = X!2D CS p res( CSp ;!) p(1!) "1! V CSp (") = V CSp (") = X n2z p 1 p(p + 1) log p "1 p 1 p(p + 1) log p D X n2z " 1 D 2 in log p (1 D 2 in log p ) cos( 2 n 2 n log p log ") i sin( log p log ") 1 D 2 in log p <(!) =D represents the amplitude of the logarithmic oscillations in the geometry of the fractal string and =(!) = 2 n log p represents the frequency.

13 p-adic self-similar strings are not Minkowski measurable p-adic Cantor strings CS p are not Minkowski measurable: doesn t exist in (0, 1) Theorem V CSp (") lim "!0 + " 1 D M All p-adic self-similar strings are lattice and lattice strings are never Minkowski measurable.

14 Average Minkowski content Average Minkowski content is the logarithmic Cesàro average: 1 M av (L p )= lim T!1 log T Z 1 1/T V Lp (") d" " 1 D " Theorem Let L p be a p-adic self-similar string of dimension D, then M av (L p )= res( L p ; D) p(1 D) M av (CS p )= p(1 1 log p+1 2 log p ) p 1 (p + 1) log p

15 Adelic Cantor strings and global complex dimensions CS CS 2 CS 3 CS 5 CS 7 A Z = R Y p<1 Z p (CS 2 CS 2 ) (CS 3 CS 3 ) (CS 5 CS 5) (CS 7 CS 7)

16 References I Appendix References M.L. Lapidus and M. van Frankenhuijsen Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (2e) Springer, R. Chugh, A. Kumar and M. Rani New 5-adic Cantor sets and fractal string SpringerPlus, a Springer Open Journal 2013 M. L. Lapidus and Nonarchimedean Cantor string and set J. Fixed Point Theory and Appl., ,

17 References II Appendix References M. L. Lapidus and Self-similar p-adic fractal strings and their complex dimensions p-adic Numbers, Ultrametric Analysis and Applications, No. 2, , M.L. Lapidus, and M. van Frankenhuijsen Minkowski measurability and exact fractal tube formulas for p-adic self-similar strings Fractal Geometry and Dynamical Systems in Pure Mathematics I: Fractals in Pure Mathematics. Contemporary Mathematics, Vol. 600, AMS 2013

18 References III Appendix References M.L. Lapidus, and M. van Frankenhuijsen Minkowski dimension and explicit tube formulas for p-adic fractal strings under review 2017 Th nk you for listening