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Nature is an infinite sphere of which the center is everywhere and the circumference nowhere. Blaise Pascal 1623-1662 L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

p -adic Cantor strings and complex fractal dimensions Michel Lapidus, L˜ u’ Hùng and Machiel van Frankenhuijsen Department of Mathematics Hawai‘i Pacific University 6Cornell Conference on Analysis, Probability and Mathematical Physics on Fractals Ithaca NY 613172017 L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Cantor set Henry John Stephen Smith discovered the Cantor set in 1874. 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 1883. Ternary Cantor set c 2 [ 0 , 1 ] : c = a 0 + a 1 3 + a 2 � C = 3 2 + · · · , a i 2 { 0 , 2 } for all i � 0 C is self-similar: C = ϕ 1 ( C ) [ ϕ 2 ( C ) where ϕ 1 ( x ) = x 3 and ϕ 1 ( x ) = x 3 + 2 3 . L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Cantor fractal string Michel Lapidus considered the complement of the Cantor set in [0,1] as an infinite sequence of lengths in 1991. 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 ∞ � 2 ζ CS ( s ) = 1 3 s + 2 3 2 s + 4 3 3 s + · · · = 1 1 � n = X 3 s � 2 3 s 3 s n = 0 Complex dimensions are poles of the geometric zeta function. They are ω = log 2 log 3 + in 2 π log 3 = D + inp , n 2 Z , for the Cantor string CS . L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Complex fractal dimensions σ CS = log 2 log 3 = D M where D M = inf { α � 0 : V CS ( ε ) = O ( ε 1 − α ) as ε ! 0 + } is the Minkowski dimension of the Cantor string and σ C S = inf { α 2 R : P ∞ n = 1 m n · l α n < 1 } is the abscissa of convergence of the Dirichlet series defining the geometric zeta function ζ CS . Theorem (M. L. Lapidus) Let L be a real fractal string with infinitely many nonzero lengths, then σ L = D M . Complex dimensions reveal oscillations intrinsic to the geometry, spectrum and dynamic of the fractal string. L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Kurt Hensel field of p -adic numbers Q p is the completion of Q wrt the p -adic norm ! [ 0 , 1 ) given by | x | p = p − v and | 0 | p = 0 . | · | p : Q � ( Q p , | · | p ) is an ultrametric space since | x + y | p  max {| x | p , | y | p } . ( Q p , | · | p ) is a nonarchimedean field since | x + x | p  | x | p . ( Q p , | · | p ) is locally compact and totally disconnected. ( R , | · | ) = ( Q ∞ , | · | ∞ ) 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! L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Alexander Ostrowski Theorem (Ostrowski Theorem) Every completion of Q is equivalent to Q p for some prime p  1 . 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 }} L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

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 { φ 1 ( x ) = 3 x , φ 2 ( x ) = 3 x + 2 } of Z 3 into itself. C 3 = { x 2 Z 3 | x = a 0 + a 1 3 + a 2 3 3 + · · · , 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 + 3 Z 3 ) [ ( 3 + 9 Z 3 ) [ ( 5 + 9 Z 3 ) [ · · · is isometric to the archimedean Cantor string CS . Complex dimensions of CS 3 are ω = log 2 log 3 + in 2 π log 3 L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

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 { φ 1 ( x ) = 5 x , φ 2 ( x ) = 5 x + 2 , φ 3 ( x ) = 5 x + 4 } . C 5 = { x 2 Z 5 | x = a 0 + a 1 5 + a 2 5 2 + · · · , 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 + 5 Z 5 ) [ ( 3 + 5 Z 5 ) [ ( 5 + 25 Z 5 ) [ ( 15 + 25 Z 5 ) [ · · · is isometric to the archimedean quinary Cantor string CS ∗ 5 . Complex dimensions of the 5-adic Cantor string CS 5 are ω = log 3 log 5 + in 2 π log 5 , n 2 Z . L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

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 { φ 1 ( x ) = px , φ 2 ( x ) = px + 2 , . . . , φ p + 1 2 ( x ) = px + p � 1 } 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 p − 1 ζ CS p ( s ) = 2 p s − p − 1 log p + 1 log p + in 2 π Complex dimensions of CS p are ω = 2 log p and p − 1 the residue of ζ CS p at ω is ( p + 1 ) log p L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Complex fractal dimensions Theorem Let L p be a p-adic fractal strings with infinitely many nonzero lengths, then σ L p = D M . Complex dimensions reveal oscillations in the geometry of p -adic fractal strings L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Exact tube formula for self-similar strings res ( ζ CS p ; ω ) X p ( 1 � ω ) ε 1 − ω V CS p ( ε ) = ω ∈ D CS p ε 1 − D − 2 π in p � 1 log p X V CS p ( ε ) = ( 1 � D � 2 π in p ( p + 1 ) log p log p ) n ∈ Z cos ( 2 π n log p log ε ) � i sin ( 2 π n log p log ε ) p � 1 p ( p + 1 ) log p ε 1 − D X V CS p ( ε ) = 1 � D � 2 π in log p n ∈ Z < ( ω ) = D represents the amplitude of the logarithmic oscillations in the geometry of the fractal string and = ( ω ) = 2 π n log p represents the frequency. L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

p -adic self-similar strings are not Minkowski measurable p -adic Cantor strings CS p are not Minkowski measurable: V CS p ( ε ) lim ε 1 − D M ε → 0 + doesn’t exist in ( 0 , 1 ) Theorem All p-adic self-similar strings are lattice and lattice strings are never Minkowski measurable. L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Average Minkowski content Average Minkowski content is the logarithmic Cesàro average: Z 1 V L p ( ε ) 1 d ε M av ( L p ) = lim log T ε 1 − D ε T →∞ 1 / T Theorem Let L p be a p-adic self-similar string of dimension D, then M av ( L p ) = res ( ζ L p ; D ) p ( 1 � D ) 1 p � 1 M av ( CS p ) = log p + 1 ( p + 1 ) log p p ( 1 � log p ) 2 L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Adelic Cantor strings and global complex dimensions Y CS ⇥ CS 2 ⇥ CS 3 ⇥ CS 5 ⇥ CS 7 ⇥ · · · ⇢ A Z = R ⇥ Z p p < ∞ ( CS 2 ⇥ CS ∗ 2 ) ⇥ ( CS 3 ⇥ CS ∗ 3 ) ⇥ ( CS 5 ⇥ CS ∗ 5 ) ⇥ ( CS 7 ⇥ CS ∗ 7 ) · · · L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Appendix References References I M.L. Lapidus and M. van Frankenhuijsen Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (2e) Springer, 2013. 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 L˜ u ’ Hùng Nonarchimedean Cantor string and set J. Fixed Point Theory and Appl. , 3 2008, 181–190. L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Appendix References References II M. L. Lapidus and L˜ u ’ Hùng Self-similar p -adic fractal strings and their complex dimensions p-Adic Numbers, Ultrametric Analysis and Applications , No. 2, 1 2009, 167–180. M.L. Lapidus, L˜ u’ Hùng 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 L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions

Appendix References References III M.L. Lapidus, L˜ u’ Hùng and M. van Frankenhuijsen Minkowski dimension and explicit tube formulas for p -adic fractal strings under review 2017 Th α nk you for listening L˜ u’ Hùng p -adic Cantor strings and complex fractal dimensions