Characterization of Host-Kra factors through a structural theorem for dynamical nilspaces
Yonatan Gutman (Institute of Mathematics, Polish Academy of Sciences)
Joint with Freddie Manners and Péter Varjú: Cubespaces and higher order Fourier analysis, 52pp. Nilspaces and nilmanifolds, 43pp. Nilspaces and inverse limit representation theorems for topological dynamical systems, 39pp.
Work in Progress: Structure theorems for Host-Kra factors of nitely generated abelian actions
Combinatorics Meets Ergodic Theory Ban, Canada
July 20-24, 2015 Szemerédi's theorem
Furstenberg proof of Szemerédi's theorem: (X , B, µ, T ) (invertible) m.p.s, f ∈ L (µ), f ≥ 0, f > 0 1 N n kn lim infN N ∑n 1 f (x)f (T x)⋯f ∞(T x)dµ(x´) > 0 Furstenberg correspondence´ principle. →∞ = 1 N n kn 2 Question: Does N ∑n 1 f (x)f (T x)⋯f (T x) converge in L (µ)? Question: Does 1 N f x f T nx f T knx converge almost N ∑n=1 ( ) ( )⋯ ( ) surely? =
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 2 Szemerédi's theorem
Furstenberg proof of Szemerédi's theorem: (X , B, µ, T ) (invertible) m.p.s, f ∈ L (µ), f ≥ 0, f > 0 1 N n kn lim infN N ∑n 1 f (x)f (T x)⋯f ∞(T x)dµ(x´) > 0 Furstenberg correspondence´ principle. →∞ = 1 N n kn 2 Question: Does N ∑n 1 f (x)f (T x)⋯f (T x) converge in L (µ)? Question: Does 1 N f x f T nx f T knx converge almost N ∑n=1 ( ) ( )⋯ ( ) surely? =
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 2 Szemerédi's theorem
Furstenberg proof of Szemerédi's theorem: (X , B, µ, T ) (invertible) m.p.s, f ∈ L (µ), f ≥ 0, f > 0 1 N n kn lim infN N ∑n 1 f (x)f (T x)⋯f ∞(T x)dµ(x´) > 0 Furstenberg correspondence´ principle. →∞ = 1 N n kn 2 Question: Does N ∑n 1 f (x)f (T x)⋯f (T x) converge in L (µ)? Question: Does 1 N f x f T nx f T knx converge almost N ∑n=1 ( ) ( )⋯ ( ) surely? =
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 2 Szemerédi's theorem
Furstenberg proof of Szemerédi's theorem: (X , B, µ, T ) (invertible) m.p.s, f ∈ L (µ), f ≥ 0, f > 0 1 N n kn lim infN N ∑n 1 f (x)f (T x)⋯f ∞(T x)dµ(x´) > 0 Furstenberg correspondence´ principle. →∞ = 1 N n kn 2 Question: Does N ∑n 1 f (x)f (T x)⋯f (T x) converge in L (µ)? Question: Does 1 N f x f T nx f T knx converge almost N ∑n=1 ( ) ( )⋯ ( ) surely? =
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 2 Szemerédi's theorem
Furstenberg proof of Szemerédi's theorem: (X , B, µ, T ) (invertible) m.p.s, f ∈ L (µ), f ≥ 0, f > 0 1 N n kn lim infN N ∑n 1 f (x)f (T x)⋯f ∞(T x)dµ(x´) > 0 Furstenberg correspondence´ principle. →∞ = 1 N n kn 2 Question: Does N ∑n 1 f (x)f (T x)⋯f (T x) converge in L (µ)? Question: Does 1 N f x f T nx f T knx converge almost N ∑n=1 ( ) ( )⋯ ( ) surely? =
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 2 Szemerédi's theorem
Furstenberg proof of Szemerédi's theorem: (X , B, µ, T ) (invertible) m.p.s, f ∈ L (µ), f ≥ 0, f > 0 1 N n kn lim infN N ∑n 1 f (x)f (T x)⋯f ∞(T x)dµ(x´) > 0 Furstenberg correspondence´ principle. →∞ = 1 N n kn 2 Question: Does N ∑n 1 f (x)f (T x)⋯f (T x) converge in L (µ)? Question: Does 1 N f x f T nx f T knx converge almost N ∑n=1 ( ) ( )⋯ ( ) surely? =
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 2 A non-conventional ergodic average
Theorem (Host & Kra 2005, Ziegler 2007)
Let (X , B, µ, T ) (invertible) m.p.s, f1, f2,..., fk ∈ L (µ) then the following 2 limit converges in L (µ): ∞
N 1 n 2n kn Q f1(T x)f2(T x)⋯fk (T x) N n 1
From now on standing= assumption: (X , B, µ, T ) is ergodic. 2 1 N n 2n kn L Key step: N ∑n 1 f1(T x)f2(T x)⋯fk (T x) → 0 if the k Gowers-Host-Kra seminorm of f = fi for some i (where ⃗n⃗ = ∑i 1 i ni ): = k 1 0 f 2 lim f T nx d =x = SSS SSSk = k Q M ( ) µ( ) ` (2` + 1) k ˆ k n `,` 0,1 ⃗⃗ →∞ Example k = 2: ⃗∈[− ] ⃗∈{ } lim 1 ` f (x)f (T n1 x)f (T n2 x)f (T n1+n2 x)d (x) `→∞ (2`+1)2 ∑n1,n2=−` µ ´ Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 3 A non-conventional ergodic average
Theorem (Host & Kra 2005, Ziegler 2007)
Let (X , B, µ, T ) (invertible) m.p.s, f1, f2,..., fk ∈ L (µ) then the following 2 limit converges in L (µ): ∞
N 1 n 2n kn Q f1(T x)f2(T x)⋯fk (T x) N n 1
From now on standing= assumption: (X , B, µ, T ) is ergodic. 2 1 N n 2n kn L Key step: N ∑n 1 f1(T x)f2(T x)⋯fk (T x) → 0 if the k Gowers-Host-Kra seminorm of f = fi for some i (where ⃗n⃗ = ∑i 1 i ni ): = k 1 0 f 2 lim f T nx d =x = SSS SSSk = k Q M ( ) µ( ) ` (2` + 1) k ˆ k n `,` 0,1 ⃗⃗ →∞ Example k = 2: ⃗∈[− ] ⃗∈{ } lim 1 ` f (x)f (T n1 x)f (T n2 x)f (T n1+n2 x)d (x) `→∞ (2`+1)2 ∑n1,n2=−` µ ´ Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 3 A non-conventional ergodic average
Theorem (Host & Kra 2005, Ziegler 2007)
Let (X , B, µ, T ) (invertible) m.p.s, f1, f2,..., fk ∈ L (µ) then the following 2 limit converges in L (µ): ∞
N 1 n 2n kn Q f1(T x)f2(T x)⋯fk (T x) N n 1
From now on standing= assumption: (X , B, µ, T ) is ergodic. 2 1 N n 2n kn L Key step: N ∑n 1 f1(T x)f2(T x)⋯fk (T x) → 0 if the k Gowers-Host-Kra seminorm of f = fi for some i (where ⃗n⃗ = ∑i 1 i ni ): = k 1 0 f 2 lim f T nx d =x = SSS SSSk = k Q M ( ) µ( ) ` (2` + 1) k ˆ k n `,` 0,1 ⃗⃗ →∞ Example k = 2: ⃗∈[− ] ⃗∈{ } lim 1 ` f (x)f (T n1 x)f (T n2 x)f (T n1+n2 x)d (x) `→∞ (2`+1)2 ∑n1,n2=−` µ ´ Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 3 A non-conventional ergodic average
Theorem (Host & Kra 2005, Ziegler 2007)
Let (X , B, µ, T ) (invertible) m.p.s, f1, f2,..., fk ∈ L (µ) then the following 2 limit converges in L (µ): ∞
N 1 n 2n kn Q f1(T x)f2(T x)⋯fk (T x) N n 1
From now on standing= assumption: (X , B, µ, T ) is ergodic. 2 1 N n 2n kn L Key step: N ∑n 1 f1(T x)f2(T x)⋯fk (T x) → 0 if the k Gowers-Host-Kra seminorm of f = fi for some i (where ⃗n⃗ = ∑i 1 i ni ): = k 1 0 f 2 lim f T nx d =x = SSS SSSk = k Q M ( ) µ( ) ` (2` + 1) k ˆ k n `,` 0,1 ⃗⃗ →∞ Example k = 2: ⃗∈[− ] ⃗∈{ } lim 1 ` f (x)f (T n1 x)f (T n2 x)f (T n1+n2 x)d (x) `→∞ (2`+1)2 ∑n1,n2=−` µ ´ Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 3 The k-cubical ergodic average
The Gowers-Host-Kra seminorm of f where G countable abelian and
{F`} is a Følner sequence.
k 1 f 2 lim f n x d x SSS SSSk = k Q M (⃗⃗. ) µ( ) ` SF S ˆ ` n F k 0,1 k ` →∞ ⃗∈ ⃗∈{ } {0, 1}k = {0, 1}k ∖ {⃗0}. The k-cubical ergodic average for f ,f L : { ⃗}⃗∈{0,1}k ∈ (µ) ( ( ∞ 1 ⃗ lim f n x k Q M ⃗(⃗⃗. ) `→∞ SF S ` n⃗∈F k ⃗∈{0,1}k ` (
2 Example: G = Z = ⟨T , S⟩, k = 2: 1 ` lim f T n1 S n2 x f T n3 S n4 x f T n1+n3 S n2+n4 x 4 Q 10( ) 01( ) 11( ) `→∞ (2 + 1) ` n1,n2,n3,n4=−`
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 4 The k-cubical ergodic average
The Gowers-Host-Kra seminorm of f where G countable abelian and
{F`} is a Følner sequence.
k 1 f 2 lim f n x d x SSS SSSk = k Q M (⃗⃗. ) µ( ) ` SF S ˆ ` n F k 0,1 k ` →∞ ⃗∈ ⃗∈{ } {0, 1}k = {0, 1}k ∖ {⃗0}. The k-cubical ergodic average for f ,f L : { ⃗}⃗∈{0,1}k ∈ (µ) ( ( ∞ 1 ⃗ lim f n x k Q M ⃗(⃗⃗. ) `→∞ SF S ` n⃗∈F k ⃗∈{0,1}k ` (
2 Example: G = Z = ⟨T , S⟩, k = 2: 1 ` lim f T n1 S n2 x f T n3 S n4 x f T n1+n3 S n2+n4 x 4 Q 10( ) 01( ) 11( ) `→∞ (2 + 1) ` n1,n2,n3,n4=−`
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 4 The k-cubical ergodic average
The Gowers-Host-Kra seminorm of f where G countable abelian and
{F`} is a Følner sequence.
k 1 f 2 lim f n x d x SSS SSSk = k Q M (⃗⃗. ) µ( ) ` SF S ˆ ` n F k 0,1 k ` →∞ ⃗∈ ⃗∈{ } {0, 1}k = {0, 1}k ∖ {⃗0}. The k-cubical ergodic average for f ,f L : { ⃗}⃗∈{0,1}k ∈ (µ) ( ( ∞ 1 ⃗ lim f n x k Q M ⃗(⃗⃗. ) `→∞ SF S ` n⃗∈F k ⃗∈{0,1}k ` (
2 Example: G = Z = ⟨T , S⟩, k = 2: 1 ` lim f T n1 S n2 x f T n3 S n4 x f T n1+n3 S n2+n4 x 4 Q 10( ) 01( ) 11( ) `→∞ (2 + 1) ` n1,n2,n3,n4=−`
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 4 The k-cubical ergodic average
The Gowers-Host-Kra seminorm of f where G countable abelian and
{F`} is a Følner sequence.
k 1 f 2 lim f n x d x SSS SSSk = k Q M (⃗⃗. ) µ( ) ` SF S ˆ ` n F k 0,1 k ` →∞ ⃗∈ ⃗∈{ } {0, 1}k = {0, 1}k ∖ {⃗0}. The k-cubical ergodic average for f ,f L : { ⃗}⃗∈{0,1}k ∈ (µ) ( ( ∞ 1 ⃗ lim f n x k Q M ⃗(⃗⃗. ) `→∞ SF S ` n⃗∈F k ⃗∈{0,1}k ` (
2 Example: G = Z = ⟨T , S⟩, k = 2: 1 ` lim f T n1 S n2 x f T n3 S n4 x f T n1+n3 S n2+n4 x 4 Q 10( ) 01( ) 11( ) `→∞ (2 + 1) ` n1,n2,n3,n4=−`
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 4 A characteristic factor for the k-cubical ergodic average
De nition
A factor (X , B, µ, G) → (Z, Z, µZ , G) is a characteristic factor for the k-cubical ergodic average if for all {f} 0 1 k ⊂ L (µ): , ( ∞ ⃗ ⃗∈{ } 1 1 L2 f n x f n x µ 0 SS k Q M (⃗⃗. ) − k Q M E( SZ)(⃗⃗. )SS2 Ð→ SF`S k k SF`S k k k n F 0,1 ( n F 0,1 ( ( ) ` ⃗ ` ⃗ →∞ ⃗∈ ⃗∈{ } ⃗∈ ⃗∈{ } ` Theorem (Host & Kra 2005, Ziegler 2007 G Z; Griesmer's Thesis 2009 G Z ; G 2015)
Let (X , B, µ, G) be an ergodic m.p.s where= G is a nitely generated= ` abelian group, e.g. G = Z , then there is a characteristic factor for the k-cubical ergodic average which has a strictly ergodic topological model (Mk , G) which is an inverse limit of k-step nilsystems given by G-equivariant continuous maps:
(Mk , G) = lim(Ln~Γn, G) ←Ð Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 5 A characteristic factor for the k-cubical ergodic average
De nition
A factor (X , B, µ, G) → (Z, Z, µZ , G) is a characteristic factor for the k-cubical ergodic average if for all {f} 0 1 k ⊂ L (µ): , ( ∞ ⃗ ⃗∈{ } 1 1 L2 f n x f n x µ 0 SS k Q M (⃗⃗. ) − k Q M E( SZ)(⃗⃗. )SS2 Ð→ SF`S k k SF`S k k k n F 0,1 ( n F 0,1 ( ( ) ` ⃗ ` ⃗ →∞ ⃗∈ ⃗∈{ } ⃗∈ ⃗∈{ } ` Theorem (Host & Kra 2005, Ziegler 2007 G Z; Griesmer's Thesis 2009 G Z ; G 2015)
Let (X , B, µ, G) be an ergodic m.p.s where= G is a nitely generated= ` abelian group, e.g. G = Z , then there is a characteristic factor for the k-cubical ergodic average which has a strictly ergodic topological model (Mk , G) which is an inverse limit of k-step nilsystems given by G-equivariant continuous maps:
(Mk , G) = lim(Ln~Γn, G) ←Ð Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 5 Nilsystems
De nition Let L be a k-step nilpotent Lie group and Γ ⊂ L a discrete cocompact subgroup. Suppose φ ∶ G → L is a continuous group homomorphism and let G act on X = L~Γ through φ. X is called a k-step nilmanifold. (G, X ) is called a k-step nilsystem.
Example (Heisenberg Nilsystem)
⎛ 1 RR ⎞ ⎛ 1 ZZ ⎞ L = ⎜ 0 1 R ⎟ ;Γ = ⎜ 0 1 Z ⎟ ⎝ 0 0 1 ⎠ ⎝ 0 0 1 ⎠
⎛ 1 α 0 ⎞ φ ∶ Z → L, 1 ↦ ⎜ 0 1 β ⎟ ⎝ 0 0 1 ⎠
where (α, β) is an irrational vector. Z ↝ X = L~Γ
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 6 Nilsystems
De nition Let L be a k-step nilpotent Lie group and Γ ⊂ L a discrete cocompact subgroup. Suppose φ ∶ G → L is a continuous group homomorphism and let G act on X = L~Γ through φ. X is called a k-step nilmanifold. (G, X ) is called a k-step nilsystem.
Example (Heisenberg Nilsystem)
⎛ 1 RR ⎞ ⎛ 1 ZZ ⎞ L = ⎜ 0 1 R ⎟ ;Γ = ⎜ 0 1 Z ⎟ ⎝ 0 0 1 ⎠ ⎝ 0 0 1 ⎠
⎛ 1 α 0 ⎞ φ ∶ Z → L, 1 ↦ ⎜ 0 1 β ⎟ ⎝ 0 0 1 ⎠
where (α, β) is an irrational vector. Z ↝ X = L~Γ
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 6 Outline of the proof
(Camarena & Szegedy, 2010) Introduce a robust topological category which captures the property of being an inverse limit of nilsystems=Nilspaces. Show by induction there is a characteristic factor for the k-cubical ergodic average which is a nilspace. The base case corresponds to the Kronecker factor.
Theorem (G, Manners & Varjú, 2014) Suppose G has a dense compactly generated subgroup. Let (G, X ) be a minimal topological dynamical system, where X is a nilspace of degree k and suppose G acts on X through a continuous group homomorphism G → Aut1(X ). Then X is homeomorphic to an inverse limit of minimal k-step nilsystems.
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 7 Outline of the proof
(Camarena & Szegedy, 2010) Introduce a robust topological category which captures the property of being an inverse limit of nilsystems=Nilspaces. Show by induction there is a characteristic factor for the k-cubical ergodic average which is a nilspace. The base case corresponds to the Kronecker factor.
Theorem (G, Manners & Varjú, 2014) Suppose G has a dense compactly generated subgroup. Let (G, X ) be a minimal topological dynamical system, where X is a nilspace of degree k and suppose G acts on X through a continuous group homomorphism G → Aut1(X ). Then X is homeomorphic to an inverse limit of minimal k-step nilsystems.
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 7 Outline of the proof
(Camarena & Szegedy, 2010) Introduce a robust topological category which captures the property of being an inverse limit of nilsystems=Nilspaces. Show by induction there is a characteristic factor for the k-cubical ergodic average which is a nilspace. The base case corresponds to the Kronecker factor.
Theorem (G, Manners & Varjú, 2014) Suppose G has a dense compactly generated subgroup. Let (G, X ) be a minimal topological dynamical system, where X is a nilspace of degree k and suppose G acts on X through a continuous group homomorphism G → Aut1(X ). Then X is homeomorphic to an inverse limit of minimal k-step nilsystems.
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 7 Outline of the proof
(Camarena & Szegedy, 2010) Introduce a robust topological category which captures the property of being an inverse limit of nilsystems=Nilspaces. Show by induction there is a characteristic factor for the k-cubical ergodic average which is a nilspace. The base case corresponds to the Kronecker factor.
Theorem (G, Manners & Varjú, 2014) Suppose G has a dense compactly generated subgroup. Let (G, X ) be a minimal topological dynamical system, where X is a nilspace of degree k and suppose G acts on X through a continuous group homomorphism G → Aut1(X ). Then X is homeomorphic to an inverse limit of minimal k-step nilsystems.
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 7 Outline of the proof
(Camarena & Szegedy, 2010) Introduce a robust topological category which captures the property of being an inverse limit of nilsystems=Nilspaces. Show by induction there is a characteristic factor for the k-cubical ergodic average which is a nilspace. The base case corresponds to the Kronecker factor.
Theorem (G, Manners & Varjú, 2014) Suppose G has a dense compactly generated subgroup. Let (G, X ) be a minimal topological dynamical system, where X is a nilspace of degree k and suppose G acts on X through a continuous group homomorphism G → Aut1(X ). Then X is homeomorphic to an inverse limit of minimal k-step nilsystems.
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 7 Labeled cubes
k X k ≜ X 0,1 [ ] { } x011 x111
x001 x101
x010 x110
x000 x100
k 0,1 k 0 k k k X ( ≜ X , π0 ∶ X → X , π( ∶ X → X ( [ ] { } ∖⃗ [ ] [ ] [ ]
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 8 Labeled cubes
k X k ≜ X 0,1 [ ] { } x011 x111
x001 x101
x010 x110
x000 x100
k 0,1 k 0 k k k X ( ≜ X , π0 ∶ X → X , π( ∶ X → X ( [ ] { } ∖⃗ [ ] [ ] [ ]
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 8 Labeled cubes
k X k ≜ X 0,1 [ ] { } x011 x111
x001 x101
x010 x110
x000 x100
k 0,1 k 0 k k k X ( ≜ X , π0 ∶ X → X , π( ∶ X → X ( [ ] { } ∖⃗ [ ] [ ] [ ]
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 8 Labeled cubes
k X k ≜ X 0,1 [ ] { } x011 x111
x001 x101
x010 x110
x000 x100
k 0,1 k 0 k k k X ( ≜ X , π0 ∶ X → X , π( ∶ X → X ( [ ] { } ∖⃗ [ ] [ ] [ ]
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 8 Labeled cubes
k X k ≜ X 0,1 [ ] { } x011 x111
x001 x101
x010 x110
x000 x100
k 0,1 k 0 k k k X ( ≜ X , π0 ∶ X → X , π( ∶ X → X ( [ ] { } ∖⃗ [ ] [ ] [ ]
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 8 Gluing of Gluable Cubes
c d d f
+ =
a b b e
c f
a e
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 9 Gluing of Gluable Cubes
c d d f
+ =
a b b e
c f
a e
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 9 Gluing of Gluable Cubes
c d d f
+ =
a b b e
c f
a e
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 9 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 Cubical measures
Key tool in the proof is a family of measures µ k , k = 0, 1, 2,... de ned on k 0 1 k k 0 1 k (X ≜ X , , B ≜ B , ) [ ] 0 1 2 [ ]De ne{ µ} =[µ]on X{. De ne} µ = µ × µ on X × X . µ will be de ned on[ ]X × X × X × X . [ ] [ ] k k Let I be the △G = {(t, t,..., t)S t ∈ G}-invariant σ-algebra of [ ] ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2k times k (X k ≜ X 0,1 , B k , µ k ). k 1 De ne[ ] µ { }to be[ ] the[ relative] independent joining of two copies of k k µ over[ I+ ].
[ ] k 1 k k k A fdµ = E( A f0SI )E( A f1SI )dµ ˆX [k+1] k 1 ˆX [k] k k 0,1 + [ + ] 0,1 0,1 [ ] k k k Let k ∈{ d} be the -ergodic∈{ } decomposition∈{ } k , then: µ = µω µ k (ω) △G µ [ ] ´ [ ] [ ] [ ] [ ] k I µ k 1 = µ k × µ k dµ (ω) ˆ ω ω k [ + ] [ ] [ ] [ ] I Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 10 The upper Host-Kra group
De nition Let F be a face of the discrete cube {0, 1}n. For g ∈ G de ne n g F ∈ G n ≜ G 0,1 : g ∈ F [ ] { } g F () = 0 ∉ F
0 g 0 g 0 g 0 g
n F n n The face group for the face F : ΓF ≜ ⟨g S g ∈ G⟩ ⊂ G . Note ΓF ≅ G. n n De ne: UHK (G) = ⟨ΓF S codim(F ) = 1, F is upper⟩. [ ] Remark: UHKn(G) is trivial on the ⃗0-vertex and therefore UHKn(G) = {0} ⊗ UHKn(G)
Yonatan Gutman(IMPAN) Characterization( of Host-Kra factors 11 The upper Host-Kra group
De nition Let F be a face of the discrete cube {0, 1}n. For g ∈ G de ne n g F ∈ G n ≜ G 0,1 : g ∈ F [ ] { } g F () = 0 ∉ F
0 g 0 g 0 g 0 g
n F n n The face group for the face F : ΓF ≜ ⟨g S g ∈ G⟩ ⊂ G . Note ΓF ≅ G. n n De ne: UHK (G) = ⟨ΓF S codim(F ) = 1, F is upper⟩. [ ] Remark: UHKn(G) is trivial on the ⃗0-vertex and therefore UHKn(G) = {0} ⊗ UHKn(G)
Yonatan Gutman(IMPAN) Characterization( of Host-Kra factors 11 The upper Host-Kra group
De nition Let F be a face of the discrete cube {0, 1}n. For g ∈ G de ne n g F ∈ G n ≜ G 0,1 : g ∈ F [ ] { } g F () = 0 ∉ F
0 g 0 g 0 g 0 g
n F n n The face group for the face F : ΓF ≜ ⟨g S g ∈ G⟩ ⊂ G . Note ΓF ≅ G. n n De ne: UHK (G) = ⟨ΓF S codim(F ) = 1, F is upper⟩. [ ] Remark: UHKn(G) is trivial on the ⃗0-vertex and therefore UHKn(G) = {0} ⊗ UHKn(G)
Yonatan Gutman(IMPAN) Characterization( of Host-Kra factors 11 The upper Host-Kra group
De nition Let F be a face of the discrete cube {0, 1}n. For g ∈ G de ne n g F ∈ G n ≜ G 0,1 : g ∈ F [ ] { } g F () = 0 ∉ F
0 g 0 g 0 g 0 g
n F n n The face group for the face F : ΓF ≜ ⟨g S g ∈ G⟩ ⊂ G . Note ΓF ≅ G. n n De ne: UHK (G) = ⟨ΓF S codim(F ) = 1, F is upper⟩. [ ] Remark: UHKn(G) is trivial on the ⃗0-vertex and therefore UHKn(G) = {0} ⊗ UHKn(G)
Yonatan Gutman(IMPAN) Characterization( of Host-Kra factors 11 The upper Host-Kra group
De nition Let F be a face of the discrete cube {0, 1}n. For g ∈ G de ne n g F ∈ G n ≜ G 0,1 : g ∈ F [ ] { } g F () = 0 ∉ F
0 g 0 g 0 g 0 g
n F n n The face group for the face F : ΓF ≜ ⟨g S g ∈ G⟩ ⊂ G . Note ΓF ≅ G. n n De ne: UHK (G) = ⟨ΓF S codim(F ) = 1, F is upper⟩. [ ] Remark: UHKn(G) is trivial on the ⃗0-vertex and therefore UHKn(G) = {0} ⊗ UHKn(G)
Yonatan Gutman(IMPAN) Characterization( of Host-Kra factors 11 The upper Host-Kra group
De nition Let F be a face of the discrete cube {0, 1}n. For g ∈ G de ne n g F ∈ G n ≜ G 0,1 : g ∈ F [ ] { } g F () = 0 ∉ F
0 g 0 g 0 g 0 g
n F n n The face group for the face F : ΓF ≜ ⟨g S g ∈ G⟩ ⊂ G . Note ΓF ≅ G. n n De ne: UHK (G) = ⟨ΓF S codim(F ) = 1, F is upper⟩. [ ] Remark: UHKn(G) is trivial on the ⃗0-vertex and therefore UHKn(G) = {0} ⊗ UHKn(G)
Yonatan Gutman(IMPAN) Characterization( of Host-Kra factors 11 The upper Host-Kra group
De nition Let F be a face of the discrete cube {0, 1}n. For g ∈ G de ne n g F ∈ G n ≜ G 0,1 : g ∈ F [ ] { } g F () = 0 ∉ F
0 g 0 g 0 g 0 g
n F n n The face group for the face F : ΓF ≜ ⟨g S g ∈ G⟩ ⊂ G . Note ΓF ≅ G. n n De ne: UHK (G) = ⟨ΓF S codim(F ) = 1, F is upper⟩. [ ] Remark: UHKn(G) is trivial on the ⃗0-vertex and therefore UHKn(G) = {0} ⊗ UHKn(G)
Yonatan Gutman(IMPAN) Characterization( of Host-Kra factors 11 Host-Kra factors
Let J (k be the σ-algebra of UHK (k (G)- invariant sets of k (X ( k ≜ X 0,1 ( , B( k ,µ ( k = (π() (µ k )) Isomorphism[ ] { joining:} [ if] φ ∶ W[ ] → X is a measure[ ] preserving ∗ isomorphism then for the graph joining λ = (Id ×φ) µ on W × X :
W × X = W × X mod λ ∗ W Lemma For every A ∈ J (k there exists B ∈ B such that
µ k (X × A) △ (B × X ( k ) = 0 [ ] [ ] De nition (k-th Host-Kra factor)
(Zk , Zk , µk , G) is the factor of (X , B, µ, G) which corresponds to the σ-algebra consisting of B ∈ B such that there exists A ∈ J (k with µ k (X × A) △ (B × X ( k ) = 0. [ ] [ ] Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 12 Host-Kra factors
Let J (k be the σ-algebra of UHK (k (G)- invariant sets of k (X ( k ≜ X 0,1 ( , B( k ,µ ( k = (π() (µ k )) Isomorphism[ ] { joining:} [ if] φ ∶ W[ ] → X is a measure[ ] preserving ∗ isomorphism then for the graph joining λ = (Id ×φ) µ on W × X :
W × X = W × X mod λ ∗ W Lemma For every A ∈ J (k there exists B ∈ B such that
µ k (X × A) △ (B × X ( k ) = 0 [ ] [ ] De nition (k-th Host-Kra factor)
(Zk , Zk , µk , G) is the factor of (X , B, µ, G) which corresponds to the σ-algebra consisting of B ∈ B such that there exists A ∈ J (k with µ k (X × A) △ (B × X ( k ) = 0. [ ] [ ] Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 12 Host-Kra factors
Let J (k be the σ-algebra of UHK (k (G)- invariant sets of k (X ( k ≜ X 0,1 ( , B( k ,µ ( k = (π() (µ k )) Isomorphism[ ] { joining:} [ if] φ ∶ W[ ] → X is a measure[ ] preserving ∗ isomorphism then for the graph joining λ = (Id ×φ) µ on W × X :
W × X = W × X mod λ ∗ W Lemma For every A ∈ J (k there exists B ∈ B such that
µ k (X × A) △ (B × X ( k ) = 0 [ ] [ ] De nition (k-th Host-Kra factor)
(Zk , Zk , µk , G) is the factor of (X , B, µ, G) which corresponds to the σ-algebra consisting of B ∈ B such that there exists A ∈ J (k with µ k (X × A) △ (B × X ( k ) = 0. [ ] [ ] Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 12 Host-Kra factors
Let J (k be the σ-algebra of UHK (k (G)- invariant sets of k (X ( k ≜ X 0,1 ( , B( k ,µ ( k = (π() (µ k )) Isomorphism[ ] { joining:} [ if] φ ∶ W[ ] → X is a measure[ ] preserving ∗ isomorphism then for the graph joining λ = (Id ×φ) µ on W × X :
W × X = W × X mod λ ∗ W Lemma For every A ∈ J (k there exists B ∈ B such that
µ k (X × A) △ (B × X ( k ) = 0 [ ] [ ] De nition (k-th Host-Kra factor)
(Zk , Zk , µk , G) is the factor of (X , B, µ, G) which corresponds to the σ-algebra consisting of B ∈ B such that there exists A ∈ J (k with µ k (X × A) △ (B × X ( k ) = 0. [ ] [ ] Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 12 Host-Kra factors
Let J (k be the σ-algebra of UHK (k (G)- invariant sets of k (X ( k ≜ X 0,1 ( , B( k ,µ ( k = (π() (µ k )) Isomorphism[ ] { joining:} [ if] φ ∶ W[ ] → X is a measure[ ] preserving ∗ isomorphism then for the graph joining λ = (Id ×φ) µ on W × X :
W × X = W × X mod λ ∗ W Lemma For every A ∈ J (k there exists B ∈ B such that
µ k (X × A) △ (B × X ( k ) = 0 [ ] [ ] De nition (k-th Host-Kra factor)
(Zk , Zk , µk , G) is the factor of (X , B, µ, G) which corresponds to the σ-algebra consisting of B ∈ B such that there exists A ∈ J (k with µ k (X × A) △ (B × X ( k ) = 0. [ ] [ ] Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 12 Structure theory for Host-Kra factors
Theorem A The extension (Zk 1, Zk 1, µk 1, G) → (Zk , Zk , µk , G) is an abelian group extension. + + +
k k k (Zk , Zk , µk ) π0 [ ] [ ] [ ] π(
S k k k (Zk , Zk , µk ) (Z , Z , µ ) k k k [ ] [ ] [ ] ( ( ( Measurable unique completion of corners isomorphism:
k k k k k k k k G ∶ (Z , Z , µ , △ ) → (Z , Z , µ , △ ) k k k G k k k G [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ( ( G (x() =∗(S(x(), x()
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 13 Structure theory for Host-Kra factors
Theorem A The extension (Zk 1, Zk 1, µk 1, G) → (Zk , Zk , µk , G) is an abelian group extension. + + +
k k k (Zk , Zk , µk ) π0 [ ] [ ] [ ] π(
S k k k (Zk , Zk , µk ) (Z , Z , µ ) k k k [ ] [ ] [ ] ( ( ( Measurable unique completion of corners isomorphism:
k k k k k k k k G ∶ (Z , Z , µ , △ ) → (Z , Z , µ , △ ) k k k G k k k G [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ( ( G (x() =∗(S(x(), x()
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 13 Structure theory for Host-Kra factors
Theorem A The extension (Zk 1, Zk 1, µk 1, G) → (Zk , Zk , µk , G) is an abelian group extension. + + +
k k k (Zk , Zk , µk ) π0 [ ] [ ] [ ] π(
S k k k (Zk , Zk , µk ) (Z , Z , µ ) k k k [ ] [ ] [ ] ( ( ( Measurable unique completion of corners isomorphism:
k k k k k k k k G ∶ (Z , Z , µ , △ ) → (Z , Z , µ , △ ) k k k G k k k G [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ( ( G (x() =∗(S(x(), x()
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 13 Structure theory for Host-Kra factors
Theorem A The extension (Zk 1, Zk 1, µk 1, G) → (Zk , Zk , µk , G) is an abelian group extension. + + +
k k k (Zk , Zk , µk ) π0 [ ] [ ] [ ] π(
S k k k (Zk , Zk , µk ) (Z , Z , µ ) k k k [ ] [ ] [ ] ( ( ( Measurable unique completion of corners isomorphism:
k k k k k k k k G ∶ (Z , Z , µ , △ ) → (Z , Z , µ , △ ) k k k G k k k G [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ( ( G (x() =∗(S(x(), x()
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 13 Structure theory for Host-Kra factors
Relative independent joining above the Host-Kra factor:
(X k , B k , µ k ) π0 [ ] [ ] [ ] π(
(X , B, µ) (X ( k , B( k , µ( k ) S [ ] [ ] [ ]
X k k k (Zk , Zk , µk ) ≃ ( ( , J ( , µ( S (k ) [ ] [ ] J
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 14 Nilspaces of degree k axioms
De nition n n X = (X , {C (X ) ⊂⊂ X }n 0) is an ergodic nilspace of degree k if the following holds: [ ] ∞ = m n n (I ) Cube Invariance: For any m, n ∈ Z , f ∈ C ({0, 1} ) and c ∈ C (X ), c ○ f ∈ C m(X ). + (E) Ergodicity: C 1(X ) = X 1 . n n ⃗ (C) Completion: For any n [∈ ]Z , if f ( ∶ {0, 1}( ≜ {0, 1} ∖ {1} → X has the property that for every 1 i n, f C n 1 X where ≤ ≤ ( () Fi ∈ ( ) n + n Fi = {x⃗ ∈ {0, 1} S xi = 0} then there exists c ∈ C (−X ) with c c f . f is referred to asS an n 1 -corner. c is refereed to ≜ 0,1 n 1 = ( ( ( − ) as∗ a completion of f (. S{ } ∖{⃗} k 1 (U)k 1 (k + 1)-Uniqueness: If h, f ∈ C (X ) and h = f then h = f . + ∗ ∗ A cubespace+ (epi)morphism between two nilspaces f ∶ X → Y consists of a n n continuous map f ∶ X → Y such that f (C (X )) ⊂ (=)C (Y ), n ∈ Z .
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors+ 15 Nilspaces of degree k axioms
De nition n n X = (X , {C (X ) ⊂⊂ X }n 0) is an ergodic nilspace of degree k if the following holds: [ ] ∞ = m n n (I ) Cube Invariance: For any m, n ∈ Z , f ∈ C ({0, 1} ) and c ∈ C (X ), c ○ f ∈ C m(X ). + (E) Ergodicity: C 1(X ) = X 1 . n n ⃗ (C) Completion: For any n [∈ ]Z , if f ( ∶ {0, 1}( ≜ {0, 1} ∖ {1} → X has the property that for every 1 i n, f C n 1 X where ≤ ≤ ( () Fi ∈ ( ) n + n Fi = {x⃗ ∈ {0, 1} S xi = 0} then there exists c ∈ C (−X ) with c c f . f is referred to asS an n 1 -corner. c is refereed to ≜ 0,1 n 1 = ( ( ( − ) as∗ a completion of f (. S{ } ∖{⃗} k 1 (U)k 1 (k + 1)-Uniqueness: If h, f ∈ C (X ) and h = f then h = f . + ∗ ∗ A cubespace+ (epi)morphism between two nilspaces f ∶ X → Y consists of a n n continuous map f ∶ X → Y such that f (C (X )) ⊂ (=)C (Y ), n ∈ Z .
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors+ 15 Nilspaces of degree k axioms
De nition n n X = (X , {C (X ) ⊂⊂ X }n 0) is an ergodic nilspace of degree k if the following holds: [ ] ∞ = m n n (I ) Cube Invariance: For any m, n ∈ Z , f ∈ C ({0, 1} ) and c ∈ C (X ), c ○ f ∈ C m(X ). + (E) Ergodicity: C 1(X ) = X 1 . n n ⃗ (C) Completion: For any n [∈ ]Z , if f ( ∶ {0, 1}( ≜ {0, 1} ∖ {1} → X has the property that for every 1 i n, f C n 1 X where ≤ ≤ ( () Fi ∈ ( ) n + n Fi = {x⃗ ∈ {0, 1} S xi = 0} then there exists c ∈ C (−X ) with c c f . f is referred to asS an n 1 -corner. c is refereed to ≜ 0,1 n 1 = ( ( ( − ) as∗ a completion of f (. S{ } ∖{⃗} k 1 (U)k 1 (k + 1)-Uniqueness: If h, f ∈ C (X ) and h = f then h = f . + ∗ ∗ A cubespace+ (epi)morphism between two nilspaces f ∶ X → Y consists of a n n continuous map f ∶ X → Y such that f (C (X )) ⊂ (=)C (Y ), n ∈ Z .
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors+ 15 Nilspaces of degree k axioms
De nition n n X = (X , {C (X ) ⊂⊂ X }n 0) is an ergodic nilspace of degree k if the following holds: [ ] ∞ = m n n (I ) Cube Invariance: For any m, n ∈ Z , f ∈ C ({0, 1} ) and c ∈ C (X ), c ○ f ∈ C m(X ). + (E) Ergodicity: C 1(X ) = X 1 . n n ⃗ (C) Completion: For any n [∈ ]Z , if f ( ∶ {0, 1}( ≜ {0, 1} ∖ {1} → X has the property that for every 1 i n, f C n 1 X where ≤ ≤ ( () Fi ∈ ( ) n + n Fi = {x⃗ ∈ {0, 1} S xi = 0} then there exists c ∈ C (−X ) with c c f . f is referred to asS an n 1 -corner. c is refereed to ≜ 0,1 n 1 = ( ( ( − ) as∗ a completion of f (. S{ } ∖{⃗} k 1 (U)k 1 (k + 1)-Uniqueness: If h, f ∈ C (X ) and h = f then h = f . + ∗ ∗ A cubespace+ (epi)morphism between two nilspaces f ∶ X → Y consists of a n n continuous map f ∶ X → Y such that f (C (X )) ⊂ (=)C (Y ), n ∈ Z .
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors+ 15 Nilspaces of degree k axioms
De nition n n X = (X , {C (X ) ⊂⊂ X }n 0) is an ergodic nilspace of degree k if the following holds: [ ] ∞ = m n n (I ) Cube Invariance: For any m, n ∈ Z , f ∈ C ({0, 1} ) and c ∈ C (X ), c ○ f ∈ C m(X ). + (E) Ergodicity: C 1(X ) = X 1 . n n ⃗ (C) Completion: For any n [∈ ]Z , if f ( ∶ {0, 1}( ≜ {0, 1} ∖ {1} → X has the property that for every 1 i n, f C n 1 X where ≤ ≤ ( () Fi ∈ ( ) n + n Fi = {x⃗ ∈ {0, 1} S xi = 0} then there exists c ∈ C (−X ) with c c f . f is referred to asS an n 1 -corner. c is refereed to ≜ 0,1 n 1 = ( ( ( − ) as∗ a completion of f (. S{ } ∖{⃗} k 1 (U)k 1 (k + 1)-Uniqueness: If h, f ∈ C (X ) and h = f then h = f . + ∗ ∗ A cubespace+ (epi)morphism between two nilspaces f ∶ X → Y consists of a n n continuous map f ∶ X → Y such that f (C (X )) ⊂ (=)C (Y ), n ∈ Z .
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors+ 15 Nilspaces of degree k axioms
De nition n n X = (X , {C (X ) ⊂⊂ X }n 0) is an ergodic nilspace of degree k if the following holds: [ ] ∞ = m n n (I ) Cube Invariance: For any m, n ∈ Z , f ∈ C ({0, 1} ) and c ∈ C (X ), c ○ f ∈ C m(X ). + (E) Ergodicity: C 1(X ) = X 1 . n n ⃗ (C) Completion: For any n [∈ ]Z , if f ( ∶ {0, 1}( ≜ {0, 1} ∖ {1} → X has the property that for every 1 i n, f C n 1 X where ≤ ≤ ( () Fi ∈ ( ) n + n Fi = {x⃗ ∈ {0, 1} S xi = 0} then there exists c ∈ C (−X ) with c c f . f is referred to asS an n 1 -corner. c is refereed to ≜ 0,1 n 1 = ( ( ( − ) as∗ a completion of f (. S{ } ∖{⃗} k 1 (U)k 1 (k + 1)-Uniqueness: If h, f ∈ C (X ) and h = f then h = f . + ∗ ∗ A cubespace+ (epi)morphism between two nilspaces f ∶ X → Y consists of a n n continuous map f ∶ X → Y such that f (C (X )) ⊂ (=)C (Y ), n ∈ Z .
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors+ 15 g Xk 1 Lπ˜(Hom(Tk 1, Xk 1), Xk 1, A) D − +i π˜− − C k 1 X C k 1 X X A L( ( k 1) ×Xk−1 ( k 1), k 1, ) + + − − − SνS Figure : g(x)(t) = ∑ k+1 (−1) ρx (t ○ ψ ),D(x)(●, ●●) = ρx (●●) − ρx (●), ν∈{0,1}( ν
Yonatan Gutman(IMPAN) Characterization of Host-Kra factors 16