Rigidity in dynamics and Mbius disjointness Mariusz Lemaczyk - - PowerPoint PPT Presentation

Rigidity in dynamics and Möbius disjointness

Mariusz Lemańczyk

Nicolaus Copernicus University, Toruń

Nombre premiers, déterminisme et pseudoaléa, CIRM Marseille, 4-8.11.2019 based on a joint work with Adam Kanigowski and Maksym Radziwiłł

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Sarnak’s conjecture

Definition (Möbius disjointness) Let X be a compact metric space and T : X → X a homeomorphism of it. One says that (X, T) is Möbius disjoint if limN→∞ 1

N

• nN f (T nx)µ(n) = 0 for all x ∈ X, f ∈ C(X).

Sarnak’s conjecture (2010) All zero topological entropy dynamical systems (X, T) are Möbius disjoint.

Remark: Möbius disjointness of (X, T) is often rephrased as Sarnak’s conjecture is satisfied for a topological system (X, T) or that the homeomorphism T is Möbius orthogonal.

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Sarnak’s conjecture

Definition (Möbius disjointness) Let X be a compact metric space and T : X → X a homeomorphism of it. One says that (X, T) is Möbius disjoint if limN→∞ 1

N

• nN f (T nx)µ(n) = 0 for all x ∈ X, f ∈ C(X).

Sarnak’s conjecture (2010) All zero topological entropy dynamical systems (X, T) are Möbius disjoint.

Remark: Möbius disjointness of (X, T) is often rephrased as Sarnak’s conjecture is satisfied for a topological system (X, T) or that the homeomorphism T is Möbius orthogonal.

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How do people show Möbius disjointness in selected classes?

µ is aperiodic (classical), that is: 1

N

• nN µ(an + b) → 0 for

each a, b 1. Check that each periodic system is Möbius disjoint; this

can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015.

Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): (fn) ⊂ C bounded such that

1 N

• nN fp1nf p2n → 0 for each different primes p1, p2

sufficiently large then

nN fnu(n) → 0 for EACH bounded

multiplicative u : N → C.

In the dynamical context (X, T) we use it for fn = f (T nx) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p1 and T p2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals:

1 M

• mM
• 1

H

• hH µ(m + h)
• → 0 when H, M → ∞ and H = o(M).

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How do people show Möbius disjointness in selected classes?

µ is aperiodic (classical), that is: 1

N

• nN µ(an + b) → 0 for

each a, b 1. Check that each periodic system is Möbius disjoint; this

can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015.

Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): (fn) ⊂ C bounded such that

1 N

• nN fp1nf p2n → 0 for each different primes p1, p2

sufficiently large then

nN fnu(n) → 0 for EACH bounded

multiplicative u : N → C.

In the dynamical context (X, T) we use it for fn = f (T nx) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p1 and T p2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals:

1 M

• mM
• 1

H

• hH µ(m + h)
• → 0 when H, M → ∞ and H = o(M).

7 / 24

How do people show Möbius disjointness in selected classes?

µ is aperiodic (classical), that is: 1

N

• nN µ(an + b) → 0 for

each a, b 1. Check that each periodic system is Möbius disjoint; this

can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015.

Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): (fn) ⊂ C bounded such that

1 N

• nN fp1nf p2n → 0 for each different primes p1, p2

sufficiently large then

nN fnu(n) → 0 for EACH bounded

multiplicative u : N → C.

In the dynamical context (X, T) we use it for fn = f (T nx) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p1 and T p2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals:

1 M

• mM
• 1

H

• hH µ(m + h)
• → 0 when H, M → ∞ and H = o(M).

7 / 24

How do people show Möbius disjointness in selected classes?

µ is aperiodic (classical), that is: 1

N

• nN µ(an + b) → 0 for

each a, b 1. Check that each periodic system is Möbius disjoint; this

can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015.

Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): (fn) ⊂ C bounded such that

1 N

• nN fp1nf p2n → 0 for each different primes p1, p2

sufficiently large then

nN fnu(n) → 0 for EACH bounded

multiplicative u : N → C.

In the dynamical context (X, T) we use it for fn = f (T nx) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p1 and T p2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals:

1 M

• mM
• 1

H

• hH µ(m + h)
• → 0 when H, M → ∞ and H = o(M).

7 / 24

How do people show Möbius disjointness in selected classes?

µ is aperiodic (classical), that is: 1

N

• nN µ(an + b) → 0 for

each a, b 1. Check that each periodic system is Möbius disjoint; this

can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015.

Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): (fn) ⊂ C bounded such that

1 N

• nN fp1nf p2n → 0 for each different primes p1, p2

sufficiently large then

nN fnu(n) → 0 for EACH bounded

multiplicative u : N → C.

In the dynamical context (X, T) we use it for fn = f (T nx) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p1 and T p2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals:

1 M

• mM
• 1

H

• hH µ(m + h)
• → 0 when H, M → ∞ and H = o(M).

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Möbius disjointness - general results

Theorem (Huang, Wang, Zhang, 2016) Möbius disjointness holds for each (X, T) for which EACH invariant measure yields a measure-theoretic system with discrete spectrum.

Short interval behaviour of µ is used. In fact, as shown later by Ferenczi, Kułaga-Przymus and L., an interpretation of a result on averaged form of Chowla conjecture1 (for the Möbius function) by Matomäki, Radziwiłł and Tao from 2015 gives that the spectral measure of the projection on the zero-coordinate in each Furstenberg system must be continuous. Discrete spectrum result immediately follows.

1

• hH
• nX

λ(n)λ(n + h)

• = o(HX)

provided that H → ∞ arbitrarily slowly with X → ∞ (and H X)

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Möbius disjointness - general results

Theorem (Huang, Wang, Zhang, 2016) Möbius disjointness holds for each (X, T) for which EACH invariant measure yields a measure-theoretic system with discrete spectrum.

Short interval behaviour of µ is used. In fact, as shown later by Ferenczi, Kułaga-Przymus and L., an interpretation of a result on averaged form of Chowla conjecture1 (for the Möbius function) by Matomäki, Radziwiłł and Tao from 2015 gives that the spectral measure of the projection on the zero-coordinate in each Furstenberg system must be continuous. Discrete spectrum result immediately follows.

1

• hH
• nX

λ(n)λ(n + h)

• = o(HX)

provided that H → ∞ arbitrarily slowly with X → ∞ (and H X)

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Möbius disjointness - general results

Theorem (Huang, Wang, Ye, 2017) Möbius disjointness holds for all systems with sub-polynomial complexity (i.e. smaller than nδ, for each δ > 0) for each invariant measure µ.

The measure complexity2 of µ is weaker than nδ if lim inf

n→∞

min{m 1 : µ(n

j=1 Bdn(xi, ε)) > 1 − ε for some x1, . . . , xm ∈ X}

nδ = 0 for each ε > 0 (here dn(y, z) = 1

n

n

j=1 d(T jy, T jz)).

All examples either with all invariant measures giving rise to discrete spectrum or C ∞-Anzai skew products. Most of the examples above fall in the category of so called rigid systems in dynamics.

2Measure complexity was introduced and studied by S. Ferenczi. 9 / 24

Möbius disjointness - general results

Theorem (Huang, Wang, Ye, 2017) Möbius disjointness holds for all systems with sub-polynomial complexity (i.e. smaller than nδ, for each δ > 0) for each invariant measure µ.

The measure complexity2 of µ is weaker than nδ if lim inf

n→∞

min{m 1 : µ(n

j=1 Bdn(xi, ε)) > 1 − ε for some x1, . . . , xm ∈ X}

nδ = 0 for each ε > 0 (here dn(y, z) = 1

n

n

j=1 d(T jy, T jz)).

All examples either with all invariant measures giving rise to discrete spectrum or C ∞-Anzai skew products. Most of the examples above fall in the category of so called rigid systems in dynamics.

2Measure complexity was introduced and studied by S. Ferenczi. 9 / 24

What about rigid systems?

(X, B, µ, T) is called rigid if T qn → Id pointwise on L2(X, B, µ). Rigidity implies zero entropy! We are interested in topological systems for which each µ ∈ M(X, T) yields a rigid system; a variant (even more ambitious), when all ergodic µ ∈ Me(X, T) yield rigid

• systems. Recall: Rigidity times depend on the measure!

Classical example (x, y) → (x, x + y) show that all ergodic members can yield rigidity, while there are invariant measures for which the spectrum can be even partly Lebesgue.

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Extending Matomäki-Radziwiłł theorem to arithmetic sequences

Theorem (Kanigowski, L., Radziwiłł, 2019) Let A > 100 and ε ∈ (0,

1 100) be given. Then, for all X > X0(ε, A),

H > H0(ε, A) and q (log X)A such that

• p|q 1/p (1 − ε)

pH 1/p,

we have

1 qXH

• a<q

2X

X

• xnx+qH

n≡a (mod q)

µ(n)

• dx ε.

Corollary For each ε ∈ (0,

1 100) there exists L0 such that for each L L0 and

q 1 satisfying

p|q 1/p (1 − ε) pL 1/p, we can find

M0 = M0(q, L) such that for all M M0, we have

M/Lq

j=0

q−1

a=0

• m∈[z+jLq,z+(j+1)Lq)

m≡a mod q

µ(m)

• εM

for some 0 z < Lq.

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Extending Matomäki-Radziwiłł theorem to arithmetic sequences

Theorem (Kanigowski, L., Radziwiłł, 2019) Let A > 100 and ε ∈ (0,

1 100) be given. Then, for all X > X0(ε, A),

H > H0(ε, A) and q (log X)A such that

• p|q 1/p (1 − ε)

pH 1/p,

we have

1 qXH

• a<q

2X

X

• xnx+qH

n≡a (mod q)

µ(n)

• dx ε.

Corollary For each ε ∈ (0,

1 100) there exists L0 such that for each L L0 and

q 1 satisfying

p|q 1/p (1 − ε) pL 1/p, we can find

M0 = M0(q, L) such that for all M M0, we have

M/Lq

j=0

q−1

a=0

• m∈[z+jLq,z+(j+1)Lq)

m≡a mod q

µ(m)

• εM

for some 0 z < Lq.

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Prime volume of a number

• p|q 1/p is called the prime volume of q.

The prime volume grows slowly with q:

• p|q

1 p log log log q + O(1). “Most” of the time, the prime volume of q stays bounded: if we set Dj :=

• q ∈ N :
• p|q

1 p < j

• ;

where obviously, Dj ⊂ Dj+1 for j ∈ N, then d(Dj) → 1 when j → ∞.

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Good rigid systems

Given (X, T) a topological dynamical system, we say that (X, T) is good if for every ν ∈ M(X, T) at least one of the following conditions holds: (BPV rigidity): (X, B, ν, T) is rigid along a sequence {qn}n1 with bounded prime volume, i.e. there exists j such that {qn}n1 ⊂ Dj; (PR rigidity): (X, B, ν, T) has polynomial rate of rigidity: there exists a linearly dense (in C(X)) set F ⊂ C(X) such that for each f ∈ F we can find δ > 0 and a sequence {qn}n1 satisfying

qδ

n

• j=−qδ

n

f ◦ T jqn − f 2

L2(ν) → 0.

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Main results

Theorem 1 (KLR, 2019) Let (X, T) be a good topological dynamical system. Then (X, T) is Möbius disjoint. Theorem 2 (KLR, 2019) Let (X, T) be a topological dynamical system. Suppose that the set of invariant ergodic measures Me(X, T) is countable. If for every ν ∈ Me(X, T) the measure theoretic dynamical system (X, B, ν, T) satisfies either BPV rigidity or PR rigidity then (X, T) is Möbius disjoint.

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Mechanism for the proof

We assume that x is generic for ν which yields a rigid system with a rigidity time {qn} along which either BPV or PR rigidity holds. Fix a continuous f : X → R, where in case PR rigidity is satisfied, we assume additionally that f ∈ F. Select Ln → ∞ slowly enough to have Ln−1

j=−Ln+1 f ◦ T jqn − f 2 L2(ν) → 0.

Note that such a sequence obviously exists when BPV rigidity (along {qn}) is satisfied, while in the case of PR rigidity, we simply take Ln = qδ

n.

Fix ε > 0 (sufficiently small). Then, for n large enough (which we fix)

• X

Ln−1

j=−Ln+1

• f ◦ T jqn − f
• 2 dν < ε.

Since x is generic for ν, we obtain limM→∞ 1

M

• mM

Ln−1

j=−Ln+1

• f (T jqn+mx) − f (T mx)
• 2

< ε. Hence, for some M0 = M0(ε) and every M > M0, we have

1 M

• mM

Ln−1

j=−Ln+1

• f (T jqn+mx) − f (T mx)
• 2

< ε. For “many” m in “many” [jLnqn, (j + 1)Lnqn]

Ln−1

j=−Ln+1

• f (T jqn+mx) − f (T mx)
• 2 < ε1/2.

For m as above, jumping by qn to the left and to the right we see a “constant” value of f : |

mM f (T mx)µ(m)| = | M/(Lnqn) j=0

• m∈[jLnqn,(j+1)Lnqn) f (T mx)µ(m)|.

Use number theory (the corollary).

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Local character of the result

The proof of Theorem 1 shows that it has a “local” character. For fixed x ∈ X and f ∈ C(X) we need to know that for each measure ν for which x is quasi-generic we have: for some {qn}, f ◦ T qn → f in L2(X, ν) and either {qn} ⊂ Dj (for some j 1) or (for some δ > 0), qδ

n

j=−qδ

n f ◦ T jqn − f 2

L2(ν) → 0

• holds. We will say that in this situation (x, f ) satisfies either

BPV or PR rigidity, respectively.

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Discussion on BPV and PR rigidity

BPV rigidity: it puts arithmetic restrictions on rigidity times; Advantage of BPV rigidity: it depends only on the measure-theoretic properties of ν, hence it does not “see” topological properties of (X, T). PR rigidity: no arithmetic restrictions on rigidity times, however topological properties of (X, T) expressed by a speed of rigidity for ν ∈ M(X, T) and certain continuous functions (namely f ∈ F) matter. Given a measure-theoretic system and BPV rigidity holds then Möbius disjointness holds in all its uniquely ergodic models. In contrast PR rigidity applies only in selected uniquely ergodic models. Consider the case of ergodic transformation with discrete spectrum. By Halmos-von Neumann theorem, it has a uniquely ergodic model in which Tx = x + a on a (metric) compact, abelian group X considered with Haar

• measure. Now, F =

X, each of characters is an eigenfunction. If f ∈ F corresponds to an eigenvalue e2πiα with α irrational then the relevant speed of rigidity is satisfied as it is ≪ qnα2 qδ

n

j=−qδ

n j2 → 0 (for a small

δ > 0), where {qn} stands for the sequence of denominators of α. If α is rational then consider just multiples of its denominators. Note that BPV rigidity is satisfied for each irrational rotation Tx = x + α

• n T (this follows from Vinogradov’s theorem on equidistribution of pα,

p =primes but it is not clear how to apply it to any ergodic discrete spectrum automorphism.

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PR rigidity versus topological rigidity

Assume that we have a homeomorphism T of a compact metric space which is pointwise rigid. We say that it satisfies the topological PR rigidity condition if for some {qn}, δ > 0 and each x ∈ X, we have

qδ

n

• j=−qδ

n

d(T jqnx, x) → 0. For F we can take the family of Lipschitz continuous functions and for each such function and each ν ∈ M(X, T), the relevant PR rigidity holds. Natural playground: smooth (area-preserving) flows on the torus.

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Are both conditions BPV and PR rigidity “stable” for all uniquely ergodic models of an ergodic automorphism?

YES if we look at Möbius disjointness!! We need to show that if PR rigidity holds in some uniquely ergodic model of T then Möbius disjointness holds in all uniquely ergodic models of T. Need to prove the strong MOMO property in ONE uniquely ergodic model (by a result of Abdalauoi, Kułaga-Przymus, L. and de la Rue, 2017).

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Strong MOMO property

Definition (El Abdalauoi, Kułaga-Przymus, L., de la Rue, 2017, Gomilko, L., de la Rue, 2019) A dynamical system (X, T) is said to satisfy the strong MOMO property if for each increasing sequence (bk) of natural numbers, bk+1 − bk → ∞, and each f ∈ C(X), we have 1 bK

• k<K
• bkn<bk+1

µ(n)f ◦ T n

• C(X)

→ 0 when K → ∞. Strong MOMO implies Möbius disjointness. Strong MOMO implies uniformity (in x ∈ X) in the definition

• f Möbius disjointness property.

Sarnak’s conjecture is equivalent to the fact that all zero entropy systems satisfy the strong MOMO property (A,K-P,L. delaR, 2017).

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PR rigidity and strong MOMO

Corollary (KLR, 2019) Assume that (Y , S) is uniquely ergodic, with the unique invariant measure ν which yields either BPV rigidity or PR rigidity. Then (Y , S) satisfies the strong MOMO property. We essentially repeat the proof of Theorem 1: by unique ergodicity, for M M0, we have 1 M

• mM

Ln

• j=−Ln

f ◦ T jqn+m − f ◦ T mC(X) < ε.

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Main corollaries

Corollary (KLR, 2019) For every d 2, almost every interval exchange transformation (IET) of d intervals is Möbius disjoint.

Chaika and Eskin in 2016 proved the above corollary for d = 3 (see also Bourgain 2012, Karagulyan 2017 for Möbius orthogonality for some subclasses of interval exchange transformations with 3 intervals), Ferenczi-Mauduit 2013 - examples of Möbius disjointness for all d 3. The proof of the corollary requires also a theorem of Chaika: For every d ∈ N and ǫ > 0 there exists 0 < a(ǫ) < 1 such that for every E ⊂ N satisfying d(E) a(ǫ), there exists a set of d IET’s of measure at least 1 − ǫ such that each IET in this set has a rigidity sequence in E. We apply Chaika’s theorem for BPV rigidity (see properties of sets Dj). We obtain not only Möbius disjointnes of these IETs but also Möbius disjointness in all uniquely ergodic models (in particular, uniform Möbius disjointness).

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Main corollaries

Corollary (KLR, 2019) For every ǫ > 0, each irrational α and φ of zero topological degree and of class C 2+ǫ, the corresponding Anzai skew product Tφ is Möbius disjoint.

T we denote the additive circle which we identify with [0, 1) with addition mod 1 Anzai skew products on T2: Tφ(x, y) := (x + α, y + φ(x)), for irrational α ∈ T and φ : T → T a continuous function on the circle. If φ is of zero topological degree, that is, φ(t) = φ(t) mod 1, where

• φ : R → R is absolutely continuous and 1-periodic (zero mean), then

T qn

φ → Id uniformly along the sequence {qn} of best rational

approximations to α (Herman, 1980th). Wang proved that all such analytic Anzai skew products are Möbius disjoint (for an earlier result see Liu-Sarnak), see also Huang-Wang-Ye for an extension of this result to the C ∞ case. Our result consists in showing that the PR condition (!) is satisfied in the C 2+ǫ-case.

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Mild mixing?

For each ε > 0, there exists H0 such that for each H H0, lim sup

N→∞

1 N

• nN
• 1

H

• hH

µ(rh + n)

• 2

ε uniformly in r 1. Corollary (hypothetic!) If the above holds then for each system (X, T) for which each invariant measure yields a rigid system we have Möbius disjointness. In particular, if T qnx → x for each x ∈ X, then (X, T) is Möbius disjoint. We have (S. Mangerel), uniformly over R: 1 R

• rR

lim sup

N→∞

1 N

• nN
• 1

H

• hH

µ(rh + n)

• 2

ε which is in fact equivalent to weak mixing (TdelaR-ML)!!

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