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 Blum1 Hank Ditton 2

1Swarthmore College 2University 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

2

Our Results 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

2

Our Results 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: f(x) = x +

∞

• n=2

anxn, with a2 = 0 (1) Conjugation by the linear map L(x) = a2x yields the following form L ◦ f ◦ L−1(x) = x + x2 + µx3 +

∞

• n=4

bnxn, with µ = a3 a2

2

(2) The above is formally equivalent to g(x) = x + x2 + µx3 (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 + x2 + µx3 Consider h3 = x + c3x3, and define c3 such that h3 ◦ f ◦ h−1

3 (x) = g(x)

mod x5 (4) Define hn(x) = x + cnxn, and for n ≥ 3 and n ∈ N Define cn such that: (hn ◦ · · · ◦ h3) ◦ f ◦

• h−1

3

• · · · ◦ h−1

n

• = g(x)

mod xn+2

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 Hn(x) = hn ◦ hn−1 ◦ · · · ◦ h3 Define H(x) = limn→∞Hn(x)

H ◦ f ◦ H−1(x) = g(x)

Since hn(x) = x + cnxn, Hn+1 = Hn mod xn+1 (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

2

Our Results 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, Qp, is the completion of Q with respect to the norm:

• m

n

• =

1 p

• rdp(m)−ordp(n)

, |0| = 0 Define ordp(m) = times p divides m

• Eg. In Q3
• 8

27

• =
• 23

33

• =

1

3

−3 = 27

• Eg. In Q2,
• 8

27

• =
• 23

33

• =

1

2

3 = 1

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 ∈ Qp 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 ⊂ Qp Qp all limits of Cauchy sequences with respect to | · |.

Note:

n an converges iff limn→∞|an| = 0

• Eg. ∞

n=0 pn ∈ Qp, since limn→∞|pn| = 0 for any p

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

2

Our Results 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 + x2 + µx3 +

∞

• n=4

anxn ∃ H(x) such that: H ◦ f ◦ H−1(x) = g(x) = x + x2 + µx3, (6) Jenkins and Spallone showed in Qp, 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 + x2 +

n=3 anxn is analytic, by assumption.

lim sup{1/| n √an|} = ǫ > 0 We can write f(x) = x + x2 + µx3 +

• n=4

bn qn xn, (7) with |bn| ≤ 1 and 0 < |q| ≤ ǫ Assume q = p. Note |p| < 1 Previous results: radius of convergence (at worst) of H(x) is |p4| = 1

p4

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 Qp, the radius of convergence can be encoded by growth of denominators. Jenkins and Spallone define σ(n) = 3n − 5 They proved H(x) = x + Anxn, with |An| ≤ 1 |(n − 2)!pσ(n)| (8) Note: |pn| ≤ |(n − 2)!| Thus, |An| ≤

1 |p4n−5| = p4n−5

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

2

Our Results 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 + x2 +

n=4 anxn

Radius of convergence of H(x) ≥ |p3| =

1 p3

In fact, consider f(x) = x + x2 +

n=k anxn, with

k ∈ N, k ≥ 4

As k → ∞, lower bound of radius of convergence of H(x) → |p2| =

1 p2

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

2

Our Results 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 + x2 +

n=k bn pn xn, for k = 4

We estimate the power of p appearing in An for all n η(n) = n − 1 + 2⌊ n−1

2 ⌋

|An| ≤

1 |(n−2)!pη(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 + 2m Let i1, . . . , iℓ ∈ N and n = ℓ

j=1 ij − ℓ + 1. Then: ℓ

• j=1

η(ij) ≤ η(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

Outline of Proof

Remember: Hn(x) = hn ◦ · · · ◦ h3(x), where hn = x + cnxn. Want to show: If |cn| ≤ |

1 (n−2)!pη(n) |, then |An| ≤ | 1 (n−2)!pη(n) |,

where Hn(x) = Anxn. From algebraic manipulation, An is a sum of terms of the form ci1 . . . ciℓ, where n =

ℓ

• j=1

ij − ℓ + 1 (9)

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

If |An| ≤

1 (n−2)!pη(n) then |cn+1| ≤ | 1 (n−1)!pη(n+1) |

From the formal algorithm: (n − 1)cn+1 = [Hn − Hn ◦ f]n+2 +

• H2

n

• n+2

Our norm is non-archimdean, so we analyze the individual terms on the right side.

Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary

Summary

For f(x) = x + x2 + ∞

n=k anxn, k ≥ 4, we improved the

minimum radius of converge of the conjugating maps

Old:

1 p4 , Our Result: 1 p3

In fact, our results generalize for functions of the form: f(x) = x + xm + ∞

n=k anxn, k ≥ 2m

Improved the method of proof Jenkins and Spallone used

Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm

The Problem Our Results Summary

Outlook

We have found functions for which we believe our estimates are sharp Consider functions of the form f(x) = x + x2 + x3

p3 + ∞ n=4 xn pn (note µ = 0)

Computer data suggests our previous results apply to these functions

Harold Blum, Hank Ditton Swarthmore College, University of Northern Colorado Local Dynamics in a P-Adic Norm