About the problem Obstruction from ergodic theory Complexity of nilsystems Complexity of nilsystems and systems lacking nilfactors Alejandro Maass (joint work with B. Host and B. Kra) University of Chile Ergodic Theory with Connections to Arithmetic, Heraklion, June 2013 Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems In studying multiple ergodic averages in a measure preserving system ( X , B , µ, T ) certain factors ( Z d : d ≥ 1) called nilfactors occur naturally ( characteristic factors ). These have counterparts in topological dynamics. If ( X , T ) is a minimal topological dynamical system ( X compact metric space and T a homeomorphism) there are analogous factors ( X d : d ≥ 1) called topological nilfactors. They play an important role in the structural analysis of this kind of systems. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems More precisely: general view of topological nilfactors (X,T) ¡ π D ¡ π d +1 π 1 ¡ π ∞ ¡ +1 ¡ . ¡. ¡. ¡. ¡ (D,T) ¡ (X d+1 ,T) ¡ (X d ,T) ¡ (X 1 ,T) ¡ (X ∞ ,T) ¡ ρ d+1,d ¡ ρ D, ∞ ¡ ρ ∞ , d +1 +1 ¡ ρ D,d +1 +1 ¡ Maximal ¡ Maximal ¡ Maximal ¡ Distal ¡Factor ¡ d+1-‑step ¡nilfactor ¡ EquiconAnuous ¡Factor ¡ Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems More precisely ... Let G be a d -step nilpotent Lie group and Γ a discrete co-compact subgroup. – Then the manifold X = G / Γ is a d-step nilmanifold . – The group G acts on X = G / Γ by left translations: for x = g Γ and τ ∈ G , τ · x = τ g Γ – A d -step nilsystem is given by ( X , B X , µ, T ) where X is a d -step nilmanifold endowed with its Haar measure µ and with the left translation by a fixed element τ ∈ G . — A nilsystem of a given order d is an inverse limit of d -step nilsystems. In this case we speak about a system of order d . Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems Example: An affine 2-step nilmanifold. 1 1 Z R Z Z G = 0 1 ; Γ := 0 1 R Z 0 0 1 0 0 1 The map 1 k β �→ ( α mod 1 , β mod 1) τ = 0 1 α 0 0 1 induces a diffeomorphism of X = G / Γ onto T 2 . The corresponding action of G on T 2 is given by: 1 k β · ( x , y ) = ( x + α, y + kx + β ) . 0 1 α 0 0 1 Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems A word about the way X d is constructed For an integer d > 1 one defines the d -regionally proximal relation denoted by RP d , which turns out to be a closed and T × T invariant equivalence relation (Host-Kra-M 2010; Shao-Ye 2013). Thus, it defines a factor of X : x RP d y if and only if x ′ , − → ∀ ǫ > 0 , ∃ y ′ , n = ( n 1 , . . . , n d ) s.t. d ( x , x ′ ) < ǫ, d ( y , y ′ ) < ǫ, d ( T η ·− → n x ′ , T η ·− → n y ′ ) < ǫ for all η ∈ { 0 , 1 } d \ { (0 , . . . , 0) } . Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems An example for d = 2 T n (y’) T m (y’) T n+m (y’) T n (x’) T m (x’) T n+m (x’) y’ x’ x y B(x, ε ) B(y, ε ) (x, y , y , y , y , y , y , y) belongs to Q 3 (x,y) belongs to RP 2 Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems Motivation: give examples of “simple” and “natural systems” with explicit, nontrivial factors Z d or X d for some d > 1. Of course, the notion of a “natural system” is not precisely defined, but it is clear that a system obtained by building an arbitrary extension of a given nilsystem is somehow “artificial”. Surprisingly, we found the task of finding non-artificial nilsystems harder than expected. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems In the ergodic setting, it turns out that for many well studied classes of systems, the factors Z d coincide with the Kronecker factor Z 1 ; because of spectral obstructions. Another relevant property seems to be some sort of topological complexity, and the concept we use is the one inspired in the notion of ε – n spanning sets. This leads to compute explicitly the topological complexity of d -step nilsystems. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems The spectrum of a nilsystem An obstruction for a system to have a non-rotational nilsystem as a factor arises from spectral conditions. Theorem (Leon Green 63’ for the case that G is connected; generalizations by Stepin 69’) Any ( X = G / Γ , B , µ, T ) ergodic nilsystem has a spectrum consisting in a discrete component and a Lebesgue component of infinite multiplicity. This plus zero entropy explain why it is difficult to build examples of simple ergodic systems admitting a true nilsystem as a measure theoretic factor. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems Nilsystems arising from non-connected groups can be quite different than those arising from connected ones. For any d > 2, any d -step ergodic nilsystem that is not a rotation admits a 2-step nilfactor that is not a rotation. Thus, we only need such a spectral result for 2-step nilsystems. So in our context we proved: Theorem Let ( X = G / Γ , µ, T ) be an ergodic 2 -step nilsystem that is not a rotation. Then L 2 ( µ ) can be written as the orthogonal sum L 2 ( µ ) = H ⊕ H ′ of two closed T-invariant subspaces such that the restriction of T to H has discrete spectrum and its restriction to H ′ has Lebesgue spectrum of infinite multiplicity. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems Corollary Let ( X , µ, T ) be an ergodic system and assume that its spectrum does not admit a Lebesgue component with infinite multiplicity. Then this system does not admit any nilsystem as a factor, other than a rotation factor. Examples: Weakly mixing systems. Systems with singular maximal spectral type. Systems with finite spectral multiplicity. This class includes: Systems of finite rank (substitution dynamical systems, linearly recurrent systems, Bratteli-Vershik systems of finite topological rank, and interval exchange transformations). Systems of local rank one or of funny rank one. Since nilsystems have zero entropy, the result also applies to Systems whose Pinsker factor belongs to one of the preceding types. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems Two applications First application: lower bounds for multiple recurrence A subset of the integers is said to be syndetic if the integers can be covered by finitely many translates of the subset. Theorem Assume that ( X , µ, T ) is an ergodic system satisfying Z d ( X ) = Z 1 ( X ) for all d > 1 , for example a system satisfying one of the properties listed after the Corollary. Let p 1 , . . . , p k be integer polynomials satisfying p i (0) = 0 for 1 ≤ i ≤ k. Then for every A ⊂ X and every ε > 0 , the set n ∈ N : µ ( A ∩ T − p 1 ( n ) A ∩ T − p 2 ( n ) A ∩ . . . ∩ T − p k ( n ) A ) > µ ( A ) k +1 − ε � � (1) is syndetic. - It is enough to consider Z 2 ( X ) = Z 1 ( X ), it is known that this implies condition in the theorem. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
About the problem Obstruction from ergodic theory Complexity of nilsystems Comments: - In (Bergelson-Host-Kra 2005) it is showed that the conclusion of the theorem does not hold for non-ergodic systems, even in the simple case of k = 2, p 1 ( n ) = n and p 2 ( n ) = 2 n . The conclusion also fails for general ergodic systems, for example for k ≥ 4 and p j ( n ) = jn for 1 ≤ j ≤ k . - On the other hand, the conclusion of the theorem holds for weakly mixing systems (Bergelson 87’). Similar lower bounds for some particular choices of polynomials are found in: Bergelson-Host-Kra 05’, Frantzikinakis-Kra 06’, Frantzikinakis 08’. Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors
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