On the Complex Dimensions of Nonarchimedean Fractal Sets Alexander M. Henderson 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals Cornell College, Ithaca, New York 17 June 2017
Outline Definitions & Notation Homogeneous measures The distance zeta function p -adic spaces Iterated function systems on Q Q p Results & Examples Self-similar sets 3-adic Cantor dust Fibonacci attractors A simple McMullen carpet analog Selected Bibliography 2 / 17
Definitions & Notation 3 / 17
Definitions & Notation: Homogeneous measures Let ( X, d, µ ) be a complete, separable metric measure space such that 0 < µ ( B ( x, r )) < ∞ for all x ∈ X and r > 0. Let A ⊆ X . Definition We say that µ is q -homogeneous on A if there is some constant M > 0 such that � q µ ( B ( x, r )) � r µ ( B ( ξ, ρ )) ≤ M ρ for all 0 < ρ < r ≤ diam ( A ), all x ∈ A , and all ξ ∈ B ( x, r ). The measure theoretic Assouad dimension of A is dim As ( A ) := inf { q ≥ 0 | µ is q -homogeneous on A } . 4 / 17
Definitions & Notation: The distance zeta function Definition Suppose that dim As ( X ) = Q and that A is a bounded subset of X . For δ > 0, define A δ := { x ∈ X | d ( x, A ) ≤ δ } . The distance zeta function associated to A is given by � d ( x, A ) s − Q d µ ( x ) ζ A ( s ) = ζ A,A δ ( s ) := A δ Under relatively mild hypotheses on A , the integral above will diverge at—but be absolutely convergent to the right of—the upper Minkowski dimension of A . Definition Suppose that ζ A ( s ) can be meromorphically extended to a (strictly) larger domain. Then the complex dimensions of A , denoted P ( A ), are the poles of this extension. That is P ( A ) := { ω ∈ C | ω is a pole of ζ A ( s ) } . 5 / 17
Definitions & Notation: p -adic spaces Let p be a fixed prime number. Definition Let r ∈ Q . The p -adic absolute value of r is given by | r | p := p − n , where n is the unique integer such that there are a, b ∈ Z relatively prime to p with r = p n a b . Definition The p -adic numbers , denoted Q p , are the metric completion of Q with respect to the metric induced by the p -adic abolute value. The p -adic integers , denoted Z p , are elements of the “dressed” unit ball in Q p , i.e. Z p := B ≤ (0 , 1) = { x ∈ Q p | | x | p ≤ 1 } . Q p is equipped with the Haar measure µ such that µ ( Z p ) = 1. 6 / 17
Definitions & Notation: p -adic spaces 7 Z 7 + 3 7 Z 7 + 4 7 Z 7 7 Z 7 + 1 7 Z 7 + 5 7 Z 7 + 6 7 / 17
Definitions & Notation: p -adic spaces Let Q ∈ N and α ∈ [1 , ∞ ). Notation On the product space Q Q p , define the equivalent metrics � Q � 1 /α | x i − y i | α d α ( x , y ) := � , p i =1 and � � � d ∞ ( x , y ) := max | x i − y i | p � 1 ≤ i ≤ Q . � Lemma For any Q ∈ N and any α ∈ [1 , ∞ ] , the product space ( Q Q p , d α , µ ) satisfies dim As ( Q Q p ) = Q, where µ is the natural product measure. 8 / 17
Definitions & Notation: Iterated function systems on Q Q p Definition A self-similar iterated function system (SSIFS) on Q Q p is a finite collection of maps { ϕ j } j ∈ J , each of which is of the form ϕ j ( x ) = p k j x + b j , p . We call p − k j the contraction ratio of ϕ j . We associate to an where k j ∈ N and b j ∈ Q Q SSIFS the map of sets � Φ( E ) := ϕ j ( E ) . j ∈ J Theorem Let Φ be as above. Then there is a unique, nonempty, compact set A ⊆ Q Q p such that Φ( A ) = A . We call A the attractor of the SSIFS. 9 / 17
Definitions & Notation: Iterated function systems on Q Q p Let { ϕ j } j ∈ J be an SSIFS. Notation Let J ∗ denote the set of all finite sequences (or “words”) with entries in J . For each J = ( j 1 , j 2 , . . . , j n ) ∈ J , define ϕ J = ϕ j n ◦ ϕ j n − 1 ◦ · · · ◦ ϕ 1 . Let ω = ( ) ∈ J ∗ denote the “empty word.” We adopt the convention that ϕ ω is the identity map, i.e. ϕ ω ( x ) = x. 10 / 17
Results & Examples 11 / 17
Results & Examples: Self-similar sets Theorem Let A be the attractor of the SSIFS { ϕ j } j ∈ J on Q Q p . Further suppose that b j ∈ Z p for each j , and that ϕ j ( Z p ) ∩ ϕ j ′ ( Z p ) = ∅ for all j � = j ′ . Then ∞ � C n p − ns , ζ A ( s ) = ζ A , Ω ι ( s ) n =0 where � d ( x, A ) s − Q d µ ( x ) , ζ A , Ω ι ( s ) = Z Q p \ Φ( Z Q p ) and C n counts the number of maps of the form ϕ J for some J ∈ J ∗ with contraction ratio p − n . 12 / 17
Results & Examples: 3 -adic Cantor dust Example 3 Z 3 + 2 ϕ 3 ϕ 4 Let { ϕ j } 4 j =1 be the SSIFS on Q 2 3 that maps Z 2 3 into the four rectangles shown to the left. Let C 2 denote the attractor of this SSIFS. 3 Z 3 + 1 We may also regard C 2 as the Cartesian prod- uct of two copies of a 3-adic Cantor set. In either case, C 2 is an analog of the ternary 3 Z 3 ϕ 1 ϕ 2 Cantor dust in R 2 . 3 Z 3 3 Z 3 + 1 3 Z 3 + 2 13 / 17
Results & Examples: 3 -adic Cantor dust Example (con’t) With respect to d ∞ , we have � = 5 d ∞ ( x, C 2 ) s − 2 d µ ( x ) = µ � Z 2 3 \ Φ( Z 2 � ζ C 2 , Ω ι ( s ) = 3 ) 9 . Z 2 3 \ Φ( Z 2 3 ) Next, observe that C n := # { J ∈ J ∗ | ϕ J ( x ) = 3 n x + b J } = 4 n . Hence � 4 ∞ ∞ � n 3 s C n 3 − ns = 5 = 5 � � ζ C 2 ( s ) = ζ C 2 , Ω ι ( s ) 3 s − 4 . 9 3 s 9 n =0 n =0 Therefore P ( C 2 ) = log(4) log(3) + ✐ 2 π Z log(3) . 14 / 17
Results & Examples: Fibonacci attractors Example Fix a prime p and define maps on Q p by ϕ 2 ( x ) = p 2 x + 1 . ϕ 1 ( x ) = px, and Let F denote the attractor of the SSIFS { ϕ 1 , ϕ 2 } . Then � log( φ ) log( p ) + ✐ 2 π Z � � − log( φ ) log( p ) + ✐ (2 π + 1) Z � ∪ P ( F ) = , log( p ) log( p ) where √ φ = 1 + 5 . 2 This recovers the result of Lapidus and L˜ u’ (2008), obtained in the setting of one-dimensional p -adic fractal strings. The current approach provides a broader context for the study of p -adic fractal strings, and avoids several technical difficulties. 15 / 17
Results & Examples: A simple McMullen carpet analog 9 Z 3 + 8 Example 3 Z 3 + 2 9 Z 3 + 5 Let A denote the attractor of the IFS shown 9 Z 3 + 2 to the left. With respect to d ∞ , 9 Z 3 + 7 3 Z 3 + 1 9 Z 3 + 4 � 3 log(2) 2 log(3) + ✐ π Z � P ( A ) = 9 Z 3 + 1 log(3) 9 Z 3 + 6 � log(4) log(3) − 1 + ✐ 2 π Z � ∪ . 3 Z 3 9 Z 3 + 3 log(3) 9 Z 3 + 0 3 Z 3 3 Z 3 + 1 3 Z 3 + 2 16 / 17
Selected Bibliography [1] Michel L. Lapidus and H` ung L˜ u’, Nonarchimedean cantor set and string , J. Fixed Point Theory and Appl. 3 (2008), no. 1, 181–190. c, and Darko ˇ [2] Michel L. Lapidus, Goran Radunovi´ Zubrini´ c, Fractal zeta functions and fractal drums , Springer, 2017. [3] Curt McMullen, The Hausdorff dimension of general Sierp´ nski carpets , Nagoya Mathematical J. 96 (1984), 1–9. 17 / 17
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