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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 =

• c 2 [0, 1] : c = a0+a1

3 +a2 32 +· · · , ai 2 {0, 2}for all i 0 C is self-similar: C = ϕ1(C) [ ϕ2(C) where ϕ1(x) = x

3 and ϕ1(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 ζCS(s) = 1 3s + 2 32s + 4 33s + · · · = 1 3s

∞

X

n=0

2 3s n = 1 3s 2 Complex dimensions are poles of the geometric zeta

• function. They are ω = log 2

log 3 + in 2π 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

σCS = log 2 log 3 = DM where DM = inf{α 0 : VCS(ε) = O(ε1−α) as ε ! 0+} is the Minkowski dimension of the Cantor string and σCS = inf{α 2 R : P∞

n=1 mn · lα n < 1} is the abscissa of

convergence of the Dirichlet series defining the geometric zeta function ζCS. Theorem (M. L. Lapidus) Let L be a real fractal string with infinitely many nonzero lengths, then σL = DM. 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

Qp is the completion of Q wrt the p-adic norm | · |p : Q ! [0, 1) given by |x|p = p−v and |0|p = 0. (Qp, | · |p) is an ultrametric space since |x + y|p  max{|x|p, |y|p}. (Qp, | · |p) is a nonarchimedean field since |x + x|p  |x|p. (Qp, | · |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 Qp for some prime p  1. Qp = {avpv + · · · + a0 + a1p + a2p2 + · · · | v 2 Z, ai 2 {0, 1, 2, 3, . . . , p 1}} The unit ball in Qp is the ring of p-adic integers Zp = {a0 + a1p + a2p2 + · · · |ai 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 C3 is the self-similar set generated by the family of similarity contraction mappings {φ1(x) = 3x, φ2(x) = 3x + 2} of Z3 into itself. C3 = {x 2 Z3 | x = a0 + a13 + a233 + · · · , ai 2 {0, 2} for all i 0} C3 is naturally homeomorphic to the ternary Cantor set C. The 3-adic Cantor string CS3 is the complement of C3 in Z3. CS3 = (1 + 3Z3) [ (3 + 9Z3) [ (5 + 9Z3) [ · · · is isometric to the archimedean Cantor string CS. Complex dimensions of CS3 are ω = log 2

log 3 + in 2π 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 C5 is the self-similar set generated by the family of similarity contraction mappings {φ1(x) = 5x, φ2(x) = 5x + 2, φ3(x) = 5x + 4}. C5 = {x 2 Z5 | x = a0 + a15 + a252 + · · · , ai 2 {0, 2, 4} for all i 0} The nonarchimedean 5-adic Cantor set C5 is homeomorphic to the archimedean quinary Cantor set C∗

5.

CS5 = (1+5Z5)[(3+5Z5)[(5+25Z5)[(15+25Z5)[· · · is isometric to the archimedean quinary Cantor string CS∗

5.

Complex dimensions of the 5-adic Cantor string CS5 are ω = log 3

log 5 + in 2π 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 Cp is the self-similar set generated by the family of similarity contraction mappings {φ1(x) = px, φ2(x) = px + 2, . . . , φ p+1

2 (x) = px + p 1}

Cp = {x 2 Zp | x = a0 + a1p + a2p2 + · · · , ai 2 {0, 2, . . . , p 1} for all i 0} The nonarchimedean p-adic Cantor set Cp is homeomorphic to the archimedean pinary Cantor set C∗

p

The p-adic Cantor string CSp is the complement of the p-adic Cantor set Cp in Zp The geometric zeta function of the p-adic Cantor string is ζCSp(s) =

p−1 2ps−p−1

Complex dimensions of CSp are ω =

log p+1

2

log p + in 2π log p and

the residue of ζCSp at ω is

p−1 (p+1) log p

L˜ u’ Hùng p-adic Cantor strings and complex fractal dimensions

Complex fractal dimensions

Theorem Let Lp be a p-adic fractal strings with infinitely many nonzero lengths, then σLp = DM. 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

VCSp(ε) = X

ω∈DCSp

res(ζCSp; ω) p(1 ω) ε1−ω VCSp(ε) = X

n∈Z

p 1 p(p + 1) log p ε1−D− 2πin

log p

(1 D 2πin

log p)

VCSp(ε) = p 1 p(p + 1) log pε1−D X

n∈Z

cos( 2πn

log p log ε) i sin( 2πn log p log ε)

1 D 2πin

log p

<(ω) = D represents the amplitude of the logarithmic

• scillations in the geometry of the fractal string and

=(ω) = 2πn

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 CSp are not Minkowski measurable: lim

ε→0+

VCSp(ε) ε1−DM 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: Mav(Lp) = lim

T→∞

1 log T Z 1

1/T

VLp(ε) ε1−D dε ε Theorem Let Lp be a p-adic self-similar string of dimension D, then Mav(Lp) = res(ζLp; D) p(1 D) Mav(CSp) = 1 p(1

log p+1

2

log p )

p 1 (p + 1) log p

L˜ u’ Hùng p-adic Cantor strings and complex fractal dimensions

Adelic Cantor strings and global complex dimensions

CS ⇥ CS2 ⇥ CS3 ⇥ CS5 ⇥ CS7 ⇥ · · · ⇢ AZ = R ⇥ Y

p<∞

Zp (CS2 ⇥ CS∗

2) ⇥ (CS3 ⇥ CS∗ 3) ⇥ (CS5 ⇥ CS∗ 5) ⇥ (CS7 ⇥ 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 Thαnk you for listening

L˜ u’ Hùng p-adic Cantor strings and complex fractal dimensions