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¨ odel, 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 + 2 b , . . . , 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 C 1 ∪ . . . ∪ C q 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 T 1 , . . . , T l 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 T 1 , . . . , T l commuting homeomorphisms of X and let G be the commutative group generated by T 1 , . . . , T l . Then ∀ ε > 0 ∃ N ∈ I N ∃ S 1 , . . . S M ∈ G ∀ z 0 ∈ X ∃ n ≤ N ∃ i ≤ M ( d ( T n 1 S i z 0 , S i z o ) < ε ∧ . . . d ( T n l S i z 0 , S i z 0 ) < ε ) . To make this fully effective, we must provide the bound N and some description of the group elements S i . 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, ◮ T i are continuous, commuting homeomorphisms of X , ◮ G T is the group of homeomorphisms generated by a finite set T = { T 1 , . . . , T l } . M for the group elements of G T that can be written as We write G T words of length < M when written as words over the generators T 1 , . . . , T l . 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 ∀ ( x n ) n ∈ I N ∃ 1 ≤ i < j ≤ γ ( ε )( d X ( x i , x j ) ≤ 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 T 1 , . . . , T l be commuting homeomorphisms of X with common modulus of uniform continuity ω T and let G be the group generated by T 1 , . . . , T l . Then for every ε > 0 there exist N , M > 0 (to be defined below) such that for every x ∈ X d ( T n simultaneously 0 < n ≤ N min min i gx , gx ) < ε for i = 1 , . . . , l. g ∈ G M 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: ◮ N 1 ( ε, γ, ω ) = M 1 ( ε, γ, ω ) = γ ( ε/ 2). ◮ ϕ k +1 ( i ) = N k ( ε k +1 , γ, ω 2 ), N i ◮ ϕ k +1 M ( i ) = 2 M k ( ε k +1 , γ, ω 2 ) + N k ( ε k +1 , γ, ω 2 ). i i i +1 = ω ϕ k N ( i )+ i · ϕ k ◮ ε k 1 = ε/ 4 and ε k M ( i ) ( ε i / 2). ◮ N k +1 ( ε, γ, ω ) = ϕ k +1 ( γ ( ε/ 2)) · γ ( ε/ 2) N ◮ M k +1 ( ε, γ, ω ) = ϕ k +1 M ( γ ( ε/ 2)) · γ ( ε/ 2). Then N = N l ( ε, γ, ω ) and M = M l ( 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 other, 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