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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 (Zd : 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 (Xd : 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) ¡

(D,T) ¡ (X∞,T) ¡ (Xd+1,T) ¡ (Xd,T) ¡ (X1,T) ¡ . ¡. ¡. ¡. ¡ πD ¡ π∞ ¡ πd+1

+1 ¡

π1 ¡ ρd+1,d ¡ ρ∞,d+1

+1 ¡

ρD,∞ ¡ ρD,d+1

+1 ¡

Maximal ¡ Distal ¡Factor ¡ Maximal ¡ d+1-­‑step ¡nilfactor ¡ Maximal ¡ 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 = gΓ and τ ∈ G, τ · x = τgΓ – A d-step nilsystem is given by (X, BX, 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. G =   1 Z R 1 R 1   ; Γ :=   1 Z Z 1 Z 1   The map τ =   1 k β 1 α 1   → (α mod 1, β mod 1) induces a diffeomorphism of X = G/Γ onto T2. The corresponding action of G on T2 is given by:   1 k β 1 α 1   · (x, y) = (x + α, y + kx + β) .

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 Xd is constructed

For an integer d > 1 one defines the d-regionally proximal relation denoted by RPd, 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 RPd y if and only if ∀ ǫ > 0, ∃ y ′, x′, − → n = (n1, . . . , nd) 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

x’ x y’ y B(x,ε) B(y,ε) Tn(x’) Tm(x’) Tn+m(x’) Tn(y’) Tm(y’) Tn+m(y’) (x, y , y , y , y , y , y , y) belongs to Q3 (x,y) belongs to RP2

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 Zd or Xd 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 Zd coincide with the Kronecker factor Z1; 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 L2(µ) can be written as the orthogonal sum L2(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 Zd(X) = Z1(X) for all d > 1, for example a system satisfying one of the properties listed after the

• Corollary. Let p1, . . . , pk be integer polynomials satisfying pi(0) = 0 for

1 ≤ i ≤ k. Then for every A ⊂ X and every ε > 0, the set

• n ∈ N: µ(A ∩ T −p1(n)A ∩ T −p2(n)A ∩ . . . ∩ T −pk (n)A) > µ(A)k+1 − ε
• (1)

is syndetic.

• It is enough to consider Z2(X) = Z1(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, p1(n) = n and p2(n) = 2n. The conclusion also fails for general ergodic systems, for example for k ≥ 4 and pj(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

About the problem Obstruction from ergodic theory Complexity of nilsystems

Second application: weighted multiple averages Theorem Let (X, T) be a uniquely ergodic topological dynamical system with invariant measure µ. Assume that (X, µ, T) satisfies Zd(X) = Z1(X) for all d > 1 (is enough for d=2) and that the projection of X onto its Kronecker factor is

• continuous. Then for any Riemann integrable function φ on X, any x ∈ X, any

system (Y , ν, S), any k ≥ 1, any functions f1, . . . , fk ∈ L∞(ν), and any integer polynomials p1, . . . , pk, the averages 1 N

N−1

• n=0

φ(T nx) · Sp1(n)f1 · Sp2(n)f2 · . . . · Spk (n)fk converge in L2(ν) as N → +∞.

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:

• The second application is a strengthening of results in Host-Kra 09’ and Chu

09.’

• In particular, this result applies for substitution dynamical systems, and more

generally for many linearly recurrent systems and systems of finite topological

• rank. In Host-Kra 09’, it was proved in the case of linear polynomials for

particular sequences, including, for example the Thue-Morse sequence.

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

Topological complexity of a nilsystem

A relevant property related to having non rotational nilfactors seems to be the topological complexity of the system: Definition Let (X, T) be a topological dynamical system and let dX be a distance on X. For ε > 0 and n ≥ 1, an ε–n spanning set for (X, T) is a finite subset {x1, . . . , xm} of X such that for every x ∈ X there exists j ∈ {1, . . . , m} such that dX(T kx, T kxj) < ε for every k ∈ {0, . . . , n − 1}. Let S(ε, n) denote the minimal cardinality of an ε–n spanning set of (X, T). We call the function S the topological complexity of the system (X, T). Clearly, this notion depends on the choice of the distance, but its asymptotic behavior when n → ∞ and ε → 0 does not.

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

This notion of complexity can be also defined by using ε–n separated set. It is strongly related to the notions of complexity of a cover and of combinatorial complexity of a subshift. A transitive system (X, T) is a rotation on a compact abelian group if and

• nly if the complexity is bounded: S(ε, n) ≤ c(ε) (Blanchard-Host-M 00’).

For general nilsystems (Dong-Donoso-M-Song-Ye 11’) gave a polynomial upper bound for S(ε, n). Missing a precise lower bound. For systems of positive topological entropy the complexity grows exponentially fast.

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

General remark on the complexity of a nilsystem

Example: Consider the Heisenberg nilmanifold X = G/Γ and the element τ ∈ G: G =   1 R R 1 R 1   ; Γ :=   1 Z Z 1 Z 1   ; τ :=   1 α γ 1 β 1   The iterates: τ n ·   1 x z 1 y 1   =   1 x + nα z + nαy + nγ + n(n−1)

2

αβ 1 y + nβ 1   The red term suggests that S(ε, n) is linear.

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

Theorem (Host,Kra,M 13’) Let (X = G/Γ, T) be a minimal d-step nilsystem for some d ≥ 2 that is not an (d − 1)-step nilsystem. Then for every ε > 0 that is sufficiently small, there exist positive constants C(ε) and C ′(ε) such that S(ε, n) of (X, T) satisfies C(ε)np ≤ S(ε, n) ≤ C ′(ε)np for every n ≥ 1 where p is the total commutator dimension of X (defined in two more slides). Moreover, p ≥ d − 1 and C(ε) → +∞ when ε → 0. More precisely, for a suitably chosen distance on X, we have C(ε) = Cε−ℓ and C ′(ε) = C ′ε−ℓ where C and C ′ are positive constants and ℓ is the dimension of X.

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

Corollaries

A direct one, Corollary Let d ≥ 1 and let (X, T) be a minimal nilsystem that is not an d-step

• nilsystem. Then

lim inf

n→+∞

1 nd S(ε, n) → ∞ as ε → 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

In the context of symbolic systems, CX(n) is the number of words of length n appearing in the subshift (X, T). Corollary Let (X, T) be a transitive subshift and assume that lim inf

n→+∞

1 n CX(n) < ∞. Then (X, T) does not admit any topological nilfactor other than rotations. Therefore, for every d ≥ 1, the topological factor Xd is equal to its topological Kronecker factor X1. More generally, if for some d ≥ 1 we have lim inf

n→+∞

1 nd CX(n) < +∞, (2) then (X, T) does not admit any nilsystem as a topological factor that is not an d-step nilsystem. Therefore, for every t ≥ d, the topological factor Xt of X is equal to Xd.

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 natural question: Some kind of converse implication to the Theorem holds ?

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

Idea of the proof

Definition: The total commutator dimension. In terms of the Lie algebra of G. p =

d−1

• ℓ=1

dim

• range
• Adτ − Id)ℓ

where exp(Adτξ) = τ · exp(ξ) · τ −1. We have p ≥ d − 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

Idea of the proof

Fix (X = G/Γ, T) a d-step nilsystem; T : X → X is a left translation by some fixed τ ∈ G. G is endowed with a right invariant distance dG and X with the quotient distance dX.

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

• I. Reduction to a local problem

In order to study the complexity of the nilsystem (X, T) we need to understand under which conditions the orbits of two distinct points x0 and x remain close from each other during a long time: dX(T kx0, T kx) < ε for 0 ≤ k < n. This is not the same problem as the study of the speed of divergence of two

• rbits for “short” times: when n is large, the finite sequence

(T kx0 : 0 ≤ k < n) is nearly uniformly distributed in X. We need an asymptotic for µ(Aε,n(x0)) where Aε,n(x0) := {x ∈ X : dX(T nx, T nx0) < ε for 0 ≤ k < n}

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

Lemma For ε sufficiently small and every x0 ∈ X, Aε,n(x0) = Wε,n · x0 where Wε,n := {g ∈ G : dG(τ kgτ −k, 1G) < ε for 0 ≤ k < n} Sketch of the proof: We write π : G → X for the natural projection, x0 = π(g0) ; x = π(gg0) ; dG(g, 1) = dX(x, x0) < ε For 0 ≤ k < n there exists a unique γk ∈ Γ with dG(τ kg0γk, τ kgg0) = dX(T kx0, T kx) < ε The problem is equivalent to showing that γk = 1G for 0 ≤ k < n.

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

We proceed by induction on d. Computations and the induction hypothesis give dG(γk, g(k)) ≤ 2ε where g(k) = [g −1, τ k]. The sequence (g(k)) is a polynomial sequence in G and thus satisfies a linear induction relation. Therefore, the sequence (γk) approximatively satisfies this

• relation. Since Γ is discrete, the sequence (γk) satisfies this relation exactly.

On the other hand, if ε is sufficiently small, then γ0 = γ1 = . . . = γd = 1G and we conclude that γk = 1G for 0 ≤ k < n.

• 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

• II. The local problem

We summarize: µ

• {x ∈ X : dX(T kx0, T kx) < ε, for 0 ≤ k < n}
• = mG(Wε,n)

where Wε,n = {g ∈ G : dG(τ kgτ −k, 1G) < ε for 0 ≤ k < n} = {exp(ξ) : ξ ∈ εWn} and Wn =

• ξ ∈ G : ||Adk

τξ|| ≤ 1 for 0 ≤ k < n

• Thus,

µ

• {x ∈ X : dX(T kx0, T kx) < ε, for 0 ≤ k < n}
• = λ(εWn)

where λ is the Lebesque measure of G. By Linear algebra, λ(εWn) is of the order of c(ε)n−p The result follows.

• 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

Thanks !!

Alejandro Maass (joint work with B. Host and B. Kra) Complexity of nilsystems and systems lacking nilfactors