Proof mining in topological dynamics Philipp Gerhardy Department of - PowerPoint PPT Presentation

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Proof mining in topological dynamics Philipp Gerhardy Department of Mathematics University of Oslo Ramsey Theory in Logic, Combinatorics and Complexity, Bertinoro 25.-30.10.2009 Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Topological Dynamics - Definitions In topological dynamics, we model the behaviour of a dynamical system by: ◮ a compact metric space ( X , d ), ◮ a self-map T : X → X (potentially a homeomorphism). We write ( X , T ) for such a dynamical system. If we have a group G of self-maps (resp. homeomorphisms) of X , we write ( X , G ). Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Topological Dynamics - Definitions We call a system ( X , G ) minimal , if there is no non-trivial subset A ⊂ X that is invariant under all actions of G . We call a subset A ⊆ X homogeneous , if there is a group G ′ commuting with G , such that ( A , G ′ ) is minimal. We call a point x ∈ X recurrent in ( X , T ), if N ( d ( T n x , x ) < ε ) . ∀ ε > 0 ∃ n ∈ I We call a point x ∈ X uniformly recurrent in ( X , T ), if N ∃ n ≤ N ( d ( T m + n x , x ) < ε ) . ∀ ε > 0 ∃ N ∈ I N ∀ m ∈ I Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Topological Dynamics - Properties of Minimal Systems Let ( X , G ) be minimal, then ◮ Every orbit { Gx } , x ∈ X is dense in X . ◮ For every ε > 0 there exists a finite set g 1 , g 2 , . . . , g m such that min 1 ≤ i ≤ m d ( x , g i y ) ≤ ε for all x , y ∈ X . For a minimal dynamical system ( X , T ), we furthermore get ◮ Every x ∈ X is uniformly recurrent. Lemma. Every dynamical system has a minimal subsystem. Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Topological Dynamics - Multiple Birkhoff Recurrence Let ( X , d ) be a Multiple Birkhoff Recurrence Theorem. compact metric space and T 1 , . . . , T l commuting homeomorphisms of X. Then there exists a point x ∈ X and a sequence n k → ∞ with T n k i x → x simultaneously for i = 1 , . . . , l. An easy corollary is: Weak Multiple Birkhoff Recurrence Theorem. Let ( X , d ) be a compact metric space and T 1 , . . . , T l comm. homeomorphisms of X. Then for every ε > 0 there exist x ∈ X , n ∈ N such that d ( T n i x , x ) < ε simultaneously for i = 1 , . . . , l. The reverse direction can be shown using compactness. Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Topological Dynamics - van der Waerden’s Theorem van der Waerden’s Theorem. For any q , k ∈ I N there is an N = N ( q , k ) such that for any q-colouring of [1 , N ] some colour contains an arithmetic progression of length k. van der Waerden follows from WMBR in the following way: ◮ Let f be a q -colouring of I N and let T be the 1-shift, then { T i f } is a compact metric space with the usual metric. ◮ Two colourings f , g with distance < 1 satisfy f (1) = g (1). ◮ A multiply recurrent point (in the weak sense) and the n ∈ I N yields an arithmetic progression. How do we compute a multiply recurrent x ∈ X and an n ∈ I N ? Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Furstenberg-Weiss’ proof Lemma. Let ( X , d ) be a compact metric space and let T : X → X be a self-map of X. Then for every ε > 0 there is an N such that d ( T n x , x ) < ε . x ∈ X and an n ∈ I Take any x 0 ∈ X and consider the sequence Proof: x 0 , Tx 0 , T 2 x 0 , . . . . By compactness two elements T i x 0 , T j x 0 with i < j are close. Let x = T i x 0 and n = j − i . ◮ The point x ∈ X and n ∈ I N are constructed explicitly – given a modulus of total boundedness. ◮ The values i , n ∈ I N are bounded uniformly in x 0 . ◮ Completeness of the space is not needed. Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Furstenberg-Weiss’ proof Lemma. Assume that for any δ < 0 and T 1 , . . . T l there exists a z ∈ X and an n > 0 such that simultaneously d ( T n i z , z ) < δ . Then for any ε > 0 and S 1 , . . . , S l +1 there exist x , y ∈ X and an m > 0 such that simultaneously d ( S m i x , y ) < ε . Define T i = S i S − 1 l +1 and let x = S − n Proof. l +1 z , y = z and m = n . This is the start of the induction step to prove multiple recurrence for any l commuting homeomorphisms. Again, we have explicit constructions for x , y ∈ X which – assuming the construction of z ∈ X is uniform – are uniform in a similar way. Again, completeness is not used. Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Furstenberg-Weiss’ proof Let A ⊆ X be homogeneous (for a group G). If for any Lemma. δ > 0 there exist u , v ∈ A and n > 0 such that d ( T n u , v ) < δ . Then for every x ∈ A and ε > 0 there is a y ∈ A and an m ∈ I N such that d ( x , T m y ) < ε . Proof. Using minimality, we obtain g 1 , . . . , g l such that 1 ≤ i ≤ l d ( g i z , z ′ ) < ε/ 2. Using continuity, we find u , v such that min d ( T n g i u , g i v ) < ε/ 2. Combine with d ( g i v , x ) < ε/ 2. This is to be applied to a suitable ( l + 1)-fold product of ( X , T i ), yielding the result simultaneously for T 1 , . . . , T l +1 . Nothing of the previous (uniform) constructions is used here! Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Furstenberg-Weiss’ proof Proof of WMBR. Induction start by lemma. For induction step, pick z 0 ∈ X . Find z 1 ∈ X such that d ( T n 1 i z 1 , z 0 ) < ε 1 . Pick small enough ε 2 > 0 (using continuity and n 1 ), find z 2 ∈ X such that d ( T n 2 i z 2 , z 1 ) < ε 2 . Construct sequence z i s.t. i < j → d ( T n j + ... + n i +1 z j , z i ) < ε/ 2. By i compactness, some z i , z j ∈ X are ε/ 2-close. Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Furstenberg-Weiss’ proof Observations: ◮ Most of the constructions in the proof are explicit. ◮ Details of the constructions are forgotten. ◮ Minimality is used to recover/replace “lost” information. Girard modified the Furstenberg-Weiss proof, using the “forgotten” constructions – in particular their uniformity – to obtain a proof that does not use minimality. Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Girard’s proof Girard proved the following variant of the Multiple Birkhoff Recurrence Theorem: WMBR, Girard’s variant. 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 ) . The key here is to explicitly construct the elements S i which will be in the group generated by T 1 , . . . , T l . Philipp Gerhardy Proof mining in topological dynamics

Introduction Multiple Birkhoff Recurrence - Proof Analysis Other proofs of van der Waerden’s Theorem Other uses of minimality Conclusions Observations ◮ Except for the constructions involving minimality, one easily make the constructions explicit, i.e. describing n and the finite set of group elements. ◮ Furstenberg and Weiss use minimality for a forward construction of a potentially infinite sequence z i . ◮ Girard uses a backwards construction of sequences of arbitrary finite length, using the constructed group elements and their uniformities. This does not need minimality. Final observation: At no point the completeness is used, the result already holds for totally bounded spaces. Philipp Gerhardy Proof mining in topological dynamics