Proof Mining in Topological Dynamics Philipp Gerhardy Department of - - PowerPoint PPT Presentation

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Proof Mining in Topological Dynamics

Philipp Gerhardy Department of Mathematics University of Oslo Joint Mathematics Meeting 2009, Washington DC, Jan 5-8 Special Session on Logic and Dynamical Systems

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Proof Mining

Proof mining: Analyzing proofs – using proof theoretic methods – to extract additional information – primarily the computational content – from even non-constructive proofs. Proof-theoretic techniques: Cut Elimination and Herbrand’s Theorem; functional interpretations (G¨

• del, Kreisel) using higher

type functionals; Kreisel’s no-counterexample interpretation.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Combinatorics and Topological Dynamics

Combinatorics: Colourings of e.g. numbers, words over a finite alphabet or finite subsets of I

• N. Establish existence of arithmetic

progressions, homogeneous words or sets, etc. Topological Dynamics: Compact metric spaces (X, d), (groups of) homeomorphisms T : X → X. Establish existence of recurrent points or other recurrence properties. Combinatorial statements have proofs using topological dynamics.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Combinatorics and Topological Dynamics

Topological Dynamics: Use of abstract functional ideas/techniques – continuity, compactness, axiom of choice/Zorn’s lemma. But: Computational content is more or less obscured. Use proof mining to “recover” or “unwind” computational content. Proof theoretic techniques guide transformation of formulas and proofs into enriched counterpart with explicit computational content.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

van der Waerden’s Theorem

Definition

An arithmetic progression of length k is a sequence of the form a, a + b, a + 2b, . . . , a + (k − 1)b for integers a, b > 0.

van der Waerden’s Theorem

For any q, k > 0 there exists an N = N(q, k) > 0 such that for any q-colouring C1 ∪ . . . ∪ Cq of [−N, N] ⊆ Z Z, one of the colours contains an arithmetic progression of length k. Question: Growth rate of function N(q, k)?

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Topological dynamics

Furstenberg and Weiss prove van der Waerden’s Theorem via the following result in topological dynamics:

Multiple Birkhoff Recurrence (Furstenberg/Weiss, 1978)

Let (X, d) be a compact metric space and T1, . . . , Tl commuting homeomorphisms of X. Then there exists a point z ∈ X such that for every ε > 0 there is an n > 0 satisfying d(T n

i z, z) ≤ ε

simultaneously for i = 1, . . . , l. q-colouring of Z Z translates into compact metric space (X, d); k-term progression into point z for k homeomorphisms.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Motivation for Proof Analysis

◮ Construction of point z yields (bound on) N(q.k). ◮ Proof of Multiple Birkhoff Recurrence Theorem does not

provide explicit construction of point z.

◮ General insight into computational content of abstract

topological or functional analytic techniques.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Girard’s variant of Multiple Recurrence

Girard proved the following variant of the Multiple Birkhoff Recurrence Theorem:

MBR, variant (Girard, 1987)

Let (X, d) be a compact metric space, let T1, . . . , Tl commuting homeomorphisms of X and let G be the commutative group generated by T1, . . . , Tl. Then ∀ε > 0∃N ∈ I N∃S1, . . . SM ∈ G∀z0 ∈ X ∃n ≤ N∃i ≤ M(d(T n

1 Siz0, Sizo) < ε ∧ . . . d(T n l Siz0, Siz0) < ε).

To make this fully effective, we must provide the bound N and some description of the group elements Si.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Proofs of MBR by Furstenberg/Weiss and Girard

To analyze Girard’s variant of MBR for general compact metric spaces and arbitrary homeomorphisms, we must make explicit the following notions:

◮ (X, d) is a compact metric space, ◮ Ti are continuous, commuting homeomorphisms of X, ◮ G T is the group of homeomorphisms generated by a finite set

T = {T1, . . . , Tl}. We write G T

M for the group elements of G T that can be written as

words of length < M when written as words over the generators T1, . . . , Tl.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Making compactness explicit

(X, d) compact metric space: (X, d) metric space, totally bounded + complete; actually only total boundedness is needed. Total boundedness: for any ε > 0 there is a number k such that among any k elements two elements are ε-close. We require a modulus of total boundedness γ: ∀ε > 0∀(xn)n∈I N∃1 ≤ i < j ≤ γ(ε)(dX(xi, xj) ≤ ε).

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Multiple Birkhoff Recurrence Theorem - effective version

Enriching and transforming the proof, one obtains the following:

Multiple Birkhoff Recurrence Theorem(effective version)

Let (X, d) be a metric space with modulus of total boundedness γ, let T1, . . . , Tl be commuting homeomorphisms of X with common modulus of uniform continuity ωT and let G be the group generated by T1, . . . , Tl. Then for every ε > 0 there exist N, M > 0 (to be defined below) such that for every x ∈ X simultaneously min

0<n≤N min g∈GM

d(T n

i gx, gx) < ε for i = 1, . . . , l.

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Multiple Birkhoff Recurrence Theorem - effective version

Define:

◮ N1(ε, γ, ω) = M1(ε, γ, ω) = γ(ε/2). ◮ ϕk+1 N

(i) = Nk(εk+1

i

, γ, ω2),

◮ ϕk+1 M (i) = 2Mk(εk+1 i

, γ, ω2) + Nk(εk+1

i

, γ, ω2).

◮ εk 1 = ε/4 and εk i+1 = ωϕk

N(i)+i·ϕk M(i)(εi/2).

◮ Nk+1(ε, γ, ω) = ϕk+1 N

(γ(ε/2)) · γ(ε/2)

◮ Mk+1(ε, γ, ω) = ϕk+1 M (γ(ε/2)) · γ(ε/2).

Then N = Nl(ε, γ, ω) and M = Ml(ε, γ, ω).

Philipp Gerhardy Proof Mining in Topological Dynamics

Introduction van der Waerden’s Theorem and Multiple Birkhoff Recurrence Proof Analysis Conclusions and Future Work

Conclusions and Future Work

◮ The extracted bounds are essentially the same as the

Ackermann bounds from van der Waerden’s combinatorial proof.

◮ Relating the combinatorial and topological concepts to each

• ther, one sees that the proofs are essentially the same too.

◮ Analyze generalizations of the Multiple Birkhoff Recurrence

Theorem.

◮ Find a topological equivalent to Shelah’s combinatorial proof. ◮ Give a full of computational interpretation of the use of

compactness/Zorn’s lemma in the Furstenberg-Weiss proof.

Philipp Gerhardy Proof Mining in Topological Dynamics