Local Dynamics in a P-Adic Norm Harold Blum 1 Hank Ditton 2 1 - PowerPoint PPT Presentation

The Problem Our Results Summary Local Dynamics in a P-Adic Norm Harold Blum 1 Hank Ditton 2 1 Swarthmore College 2 University of Northern Colorado July 27,2010 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Acknowledgements We would like to thank Adrian Jenkins, our mentor Stephen Spallone K-State Mathematics Department The following research was done at an REU funded by the NSF under DMS-1004336 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Outline 1 The Problem The Basic Problem What are p-adic Numbers? Previous Results Our Results 2 Main Results Basic Ideas for Proofs Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary The Basic Problem Outline 1 The Problem The Basic Problem What are p-adic Numbers? Previous Results Our Results 2 Main Results Basic Ideas for Proofs Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary The Basic Problem Conjugacy Classes We are interested in the local conjugacy classes of analytic functions of the form: ∞ � a n x n , with a 2 � = 0 f ( x ) = x + (1) n = 2 Conjugation by the linear map L ( x ) = a 2 x yields the following form ∞ b n x n , with µ = a 3 L ◦ f ◦ L − 1 ( x ) = x + x 2 + µ x 3 + � (2) a 2 2 n = 4 The above is formally equivalent to g ( x ) = x + x 2 + µ x 3 (3) Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary The Basic Problem Formal Algorithm We construct a formal power series H ( x ) such that H ◦ f ◦ H − 1 ( x ) = g ( x ) = x + x 2 + µ x 3 Consider h 3 = x + c 3 x 3 , and define c 3 such that h 3 ◦ f ◦ h − 1 mod x 5 3 ( x ) = g ( x ) (4) Define h n ( x ) = x + c n x n , and for n ≥ 3 and n ∈ N Define c n such that: � � h − 1 ◦ · · · ◦ h − 1 mod x n + 2 ( h n ◦ · · · ◦ h 3 ) ◦ f ◦ = g ( x ) n 3 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary The Basic Problem Formal Algorithm Continued Let H n ( x ) = h n ◦ h n − 1 ◦ · · · ◦ h 3 Define H ( x ) = lim n →∞ H n ( x ) H ◦ f ◦ H − 1 ( x ) = g ( x ) Since h n ( x ) = x + c n x n , mod x n + 1 H n + 1 = H n (5) f ( x ) and g ( x ) are formally equivalent. Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary What are p-adic Numbers? Outline 1 The Problem The Basic Problem What are p-adic Numbers? Previous Results Our Results 2 Main Results Basic Ideas for Proofs Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary What are p-adic Numbers? What are the p-adic Numbers? The set of p-adic numbers, Q p , is the completion of Q with respect to the norm: � ord p ( m ) − ord p ( n ) � m � 1 � � � = , | 0 | = 0 � � n p Define ord p ( m ) = times p divides m � 1 � − 3 = 27 � � � 2 3 � = � 8 � � Eg. In Q 3 � = � � 27 3 3 3 � 1 � 3 = 1 � � � 2 3 � = � � 8 � Eg. In Q 2 , � = � � 27 3 3 2 8 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary What are p-adic Numbers? Properties of | · | The p-adic norm satisfies the properties of a non-archmidean norm, If a , b ∈ Q p then: | ab | = | a || b | | a | ≥ 0, and | a | = 0 iff a = 0 Strong triangle inequality: | a + b | ≤ max ( | a | , | b | ) Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary What are p-adic Numbers? Examples of p-adic Numbers Q ⊂ Q p Q p all limits of Cauchy sequences with respect to | · | . Note: � n a n converges iff lim n →∞ | a n | = 0 n = 0 p n ∈ Q p , since lim n →∞ | p n | = 0 for any p Eg. � ∞ Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary What are p-adic Numbers? Why study p-adic Numbers? Lefschetz’s Principle: Interesting problems in R or C have interesting analogs in the p-adics. Representation Theory Quadratic Forms Elliptic Curves Dynamics Problems can be easier to understand in the p-adics. Theory is known in C , but is difficult to link formal and analytic theory. Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Previous Results Outline 1 The Problem The Basic Problem What are p-adic Numbers? Previous Results Our Results 2 Main Results Basic Ideas for Proofs Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Previous Results Analytic Equivalence Consider ∞ f ( x ) = x + x 2 + µ x 3 + � a n x n n = 4 ∃ H ( x ) such that: H ◦ f ◦ H − 1 ( x ) = g ( x ) = x + x 2 + µ x 3 , (6) Jenkins and Spallone showed in Q p , H ( x ) is analytic. f ( x ) and g ( x ) are analytically equivalent Theory applies in any non-archimedian field Char 0 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Previous Results Analytic Equivalence f ( x ) = x + x 2 + � n = 3 a n x n is analytic, by assumption. √ a n |} = ǫ > 0 lim sup { 1 / | n We can write b n f ( x ) = x + x 2 + µ x 3 + � q n x n , (7) n = 4 with | b n | ≤ 1 and 0 < | q | ≤ ǫ Assume q = p . Note | p | < 1 Previous results: radius of convergence (at worst) of H ( x ) is | p 4 | = 1 p 4 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Previous Results The σ -function In Q p , the radius of convergence can be encoded by growth of denominators. Jenkins and Spallone define σ ( n ) = 3 n − 5 They proved H ( x ) = x + � A n x n , with 1 | A n | ≤ (8) | ( n − 2 )! p σ ( n ) | Note: | p n | ≤ | ( n − 2 )! | | p 4 n − 5 | = p 4 n − 5 1 Thus, | A n | ≤ Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Main Results Outline 1 The Problem The Basic Problem What are p-adic Numbers? Previous Results Our Results 2 Main Results Basic Ideas for Proofs Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Main Results µ = 0 Consider µ = 0, then f ( x ) = x + x 2 + � n = 4 a n x n 1 Radius of convergence of H ( x ) ≥ | p 3 | = p 3 In fact, consider f ( x ) = x + x 2 + � n = k a n x n , with k ∈ N , k ≥ 4 As k → ∞ , lower bound of radius of convergence of H ( x ) → | p 2 | = 1 p 2 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Basic Ideas for Proofs Outline 1 The Problem The Basic Problem What are p-adic Numbers? Previous Results Our Results 2 Main Results Basic Ideas for Proofs Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Basic Ideas for Proofs The η function f ( x ) = x + x 2 + � b n p n x n , for k = 4 n = k We estimate the power of p appearing in A n for all n η ( n ) = n − 1 + 2 ⌊ n − 1 2 ⌋ 1 | A n | ≤ | ( n − 2 )! p η ( n ) | for all n ≥ 3 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Basic Ideas for Proofs η ( n ) vs. σ ( n ) Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Basic Ideas for Proofs Properties of η ( n ) = n − 1 + 2 ⌊ n − 1 2 ⌋ Three properties of η ( n ) : η ( n ) is strictly increasing and integer valued If a , b , m ∈ N and b − a ≥ m ( k − 2 ) , then η ( b ) − η ( a ) ≥ b − a + 2 m Let i 1 , . . . , i ℓ ∈ N and n = � ℓ j = 1 i j − ℓ + 1. Then: ℓ � η ( i j ) ≤ η ( n ) j = 1 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary Basic Ideas for Proofs Outline of Proof Remember: H n ( x ) = h n ◦ · · · ◦ h 3 ( x ) , where h n = x + c n x n . 1 1 Want to show: If | c n | ≤ | ( n − 2 )! p η ( n ) | , then | A n | ≤ | ( n − 2 )! p η ( n ) | , where H n ( x ) = � A n x n . From algebraic manipulation, A n is a sum of terms of the form ℓ � c i 1 . . . c i ℓ , where n = i j − ℓ + 1 (9) j = 1 Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm