Posted by: yglima | February 27, 2010

ERT8: Weak Mixing Systems

1. The dichotomy between structure and randomness

The main tool used in ERT1 and ERT2 was: given a mps {\mathbb X=(X,\mathcal A,\mu,T)}, we decomposed the space {L^2(\mu)} into two pieces: one structured, formed by the fixed (or periodic, depending on the case) functions, and other random, formed by the functions for which the Cesàro averages converge to zero. This represents an example of the dichotomy that surrounds Ergodic Ramsey Theory: structure vs. randomness. This is actually briefly discussed in the end of ERT2 and in a broader way in this paper of Terence Tao.

This idea is also used in other branches of Mathematics, specially in combinatorics, harmonic analysis, number theory, etc. We can cite many situations:

  1. Theorems about the existence of ergodic averages.
  2. Szemerédi regularity lemma.
  3. Roth theorem on the existence of arithmetic progressions of length three in sets of positive density.
  4. Gowers norms.
  5. All proofs (up to my knowledge) of Szemerédi theorem.
  6. Green-Tao theorem on the existence of arithmetic progressions in the primes.
  7. The Julia set of holomorphic functions {f:\mathbb C\rightarrow\mathbb C} is either connected or a Cantor set.
  8. Mané-Bochi ’02: a {C^1}-generic conservative diffeomorphim in surfaces either has all Lyapunov exponents zero almost everywhere or is Anosov.
  9. Avila-Moreira ’03: For almost every {a\in[-1/4,2]}, the quadratic function {f_a(x)=a-x^2} is either regular (has a periodic attractor) or stochastic (has an invariant absolutely continuous probability with positive Lyapunov exponent).
  10. Avila-Forni ’07: almost every interval exchange transformation is either an irrational rotation or weak mixing.
  11. Avila ’10: for Schrodinger operators with a one-frequency and typical real analytic potential, the spectrum is either subcritical or supercritical.

Each situation has a notion of structure/randomness. The one we are interested is multiple recurrence of mps. Let us see this from the spectral theory point of view.

Given a mps {\mathbb X=(X,\mathcal A,\mu,T)}, denote also by {T:L^2(\mu)\rightarrow L^2(\mu)} the Koopman-von Neumann operator, defined by

\displaystyle Tf=f\circ T.

When necessary, we use the notation {U_T} to denote this operator. Many of its spectral properties are related to ergodic properties of {\mathbb X}. We investigate the eigenvalues/eigenfunctions of {T}. If the eigenfunctions form a basis of {L^2(\mu)}, then {T} is determined. In fact, let {\sigma(T)\subset\mathbb S^1} be the multiset (repeated with multiplicity) of eigenvalues of {T} and, for each {\lambda\in\sigma(T)}, {f_\lambda\in L^2(\mu)} the eigenfunction associated to {\lambda}. If \displaystyle \overline{\langle \{f_\lambda\,;\,\lambda\in\sigma(T)\}\rangle}=L^2(\mu),


\displaystyle T\left(\sum a_\lambda f_\lambda\right)=\sum(a_\lambda\cdot\lambda)f_\lambda.

In this case, we say that {T} has pure point spectrum and {\mathbb X} is a compact system. This constitutes the structured notion we were looking for.

In contrast, when {T} has no eigenvalues other than {1} and it is simple, we say that {T} has continuous spectrum and {\mathbb X} is a weak mixing system. It forms the random part.

As pure point/continuous spectrum are opposite notions, there is a hope that every mps can be decomposed into two parts: one compact and other weak mixing. This is not true at all. Instead, {\mathbb X} can be decomposed in several parts in such a way that every part is an extension of the previous one and it is a compact or weak mixing extension of the smaller one. In other words, the dynamics of {\mathbb X} is broken into many parts in which every braking is obtained from the previous one by adding one of the two dynamical prototypes we discussed above.

Formally speaking, given two mps {\mathbb X=(X,\mathcal A,\mu,T)} and {\mathbb Y=(Y,\mathcal B,\nu,S)}, we say that {\mathbb X} is an extension of {\mathbb Y} if there is a surjective measurable map {\pi:(X,\mathcal A,\mu)\rightarrow(Y,\mathcal B,\nu)} such that

\displaystyle \pi\circ T=S\circ\pi.

We denote this by {\mathbb X\stackrel{\pi}\rightarrow\mathbb Y} and {\mathbb Y} is called a factor of {\mathbb X}.

Theorem 1 (Furstenberg structural theorem) Given a mps {\mathbb X=(X,\mathcal A,\mu,T)}, there exists an ordinal {\alpha} and a family of factors {\mathbb X_\beta=(X_{\beta},\mathcal{A}_{\beta},\nu_{\beta},T_{\beta})} of {\mathbb X}, for every {\beta\leq\alpha}, such that:

  • {\mathbb X_{0}} is a single point.
  • {\mathbb X_{\beta+1}} is a compact extension of {\mathbb X_{\beta}} for every successor ordinal {\beta+1\leq\alpha}.
  • {\mathbb X_{\beta}} is the inverse limit of {(\mathbb X_{\gamma})_{\gamma<\beta}}, for every limit ordinal {\beta}.
  • {\mathbb X} is a weak mixing extension of {\mathbb X_{\alpha}}.

Above, inverse limit is in the sense that {L^2(\mathbb \nu_{\beta})=\overline{\cup_{\gamma<\beta}L^2(\mathbb \nu_{\gamma})}}. This result will be discussed in Lebesgue-full detail in the last post. In order to understand it, we need to study four concepts:

  1. Weak mixing systems.
  2. Compact systems.
  3. Weak mixing extensions.
  4. Compact extensions.

These will be the topics of this and the next 2 or 3 lectures.

2. Weak mixing systems

The definition used below is different from the one we assumed above, but don’t worry: they will be shown to coincide. Actually, we will obtain various equivalent definitions of weak mixing.

Definition 2 A mps {\mathbb X=(X,\mathcal A,\mu,T)} is weak mixing if, for every {A,B\in\mathcal A}, \displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|\mu(T^{-n}A\cap B)-\mu(A)\cdot\mu(B)\right|=0\,. \ \ \ \ \ (1)

Exercise 1 Consider a bounded sequence {(a_n)_{n\ge 1}} of nonnegative real numbers. Prove:

  1. If {\lim a_n=a}, then \displaystyle \lim \dfrac{1}{N}\sum_{n=1}^{N}|a_n-a|=0. Conclude that strong mixing implies weak mixing.
  2. If {\lim N^{-1}\cdot\sum_{n=1}^{N}|a_n-a|=0}, then \displaystyle \lim\dfrac{1}{N}\sum_{n=1}^{N}a_n=a. Conclude that weak mixing implies ergodicity.

Exercise 2 {\mathbb X} is weak mixing if and only if \displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|\left(f\circ T^n,g\right)-(f,1)\cdot(1,g)\right|=0\,, \ \ \ \ \ (2)

for every {f,g\in L^2(\mu)}.

Proposition 3 {\mathbb X} is weak mixing if and only if \displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|\left(f\circ T^n,f\right)\right|=0\,, \ \ \ \ \ (3)

for every {f\in L^2(\mu)} such that {\int_X fd\mu=0}.

We leave the proof to the reader, which may be found in the book Topics in ergodic theory of W. Parry. The notion of weak mixing means that, in some sense, almost all the system behaves in a strong mixing way. This is what says the following lemma.

Lemma 4 Consider a bounded sequence {(a_n)_{n\ge 1}} of nonnegative real numbers. Then\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N a_n=0

if and only if there exists a set {E\subset{\mathbb Z}} of zero density such that\displaystyle \lim_{n\rightarrow+\infty\atop{n\not\in E}}a_n=0\,.

Proof: ({\Longrightarrow}) Define, for each {m\ge 1}, the set

\displaystyle E_m=\left\{n\in\mathbb Z\,;\,a_n\ge 1/m\right\}.

{(E_m)} is an ascending chain of subsets of {{\mathbb Z}}. They are the sets that may give problems in the convergence of {(a_n)} to zero. Each of them has zero density, because

\displaystyle \frac{1}{N}\sum_{n=1}^{N}a_n\ge\frac{1}{N}\sum_{n=1\atop{n\in E_m}}^{N}a_n\ge\frac{1}{m}\cdot\frac{|E_m\cap\{1,\ldots,N\}|}{N}

and then

\displaystyle 0\le\lim_{N\rightarrow\infty}\frac{|E_m\cap\{1,\ldots,N\}|}{N}\le m\cdot\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}a_n=0\,.

In this way, take an increasing sequence {n_1<n_2<\cdots} of integers such that \dfrac{1}{N}\sum_{n=1}^N\chi_{E_m}(n)\le\dfrac{1}{m}\,,\,\forall\,N\ge n_m. Define

\displaystyle E=\bigcup_{m\ge 1}E_m\cap[n_m,n_{m+1}).

By definition,

\displaystyle \lim_{n\rightarrow\infty\atop{n\notin E}}a_n=0\,.

It remains to prove that {E} has zero density. Consider an integer {N}, let us say, with {n_m\le N<n_{m+1}}. As {E_1\subseteq E_2\subseteq\cdots},

\displaystyle E\cap[n_i,n_{i+1})=E_i\cap[n_i,n_{i+1})\subseteq E_m\cap[n_i,n_{i+1}),\ \ \forall\,i\le m,

and then

\displaystyle E\cap\{1,\ldots,N\}\subseteq E_m\cap\{1,\ldots,N\}\,,

which, by (2), implies that


Then, {E} has zero density.

({\Longleftarrow}) Let {M>0} such that {0\le a_n\le M}, for every {n\ge 0}. Given {\varepsilon>0}, we want to prove that

\displaystyle \frac{1}{N}\sum_{n=1}^{N}a_n<\varepsilon\,,

for every {N} large enough. By hypothesis, there is {n_0>0} for which

\displaystyle \frac{1}{N}\sum_{n=1}^{N}\chi_E(n)<\frac{\varepsilon}{3M}\,,\ \ \forall\,N\ge n_0


\displaystyle a_n<\frac{\varepsilon}{3}\,,\ \ \forall\,n\notin E,\,n\ge n_0.

Then, for {N\ge n_0},

\displaystyle \begin{array}{rcl} \frac{1}{N}\sum_{n=1}^{N}a_n&=&\frac{1}{N}\sum_{n=1}^{n_0}a_n+\frac{1}{N}\sum_{n=n_0+1\atop{n\in E}}^{N}a_n+ \frac{1}{N}\sum_{k=n_0+1\atop{n\notin E}}^{N}a_n\\ &<&\frac{1}{N}\sum_{n=1}^{n_0}a_n+\frac{M}{N}\sum_{n=1}^{N}\chi_E(n)+\frac{\varepsilon}{3}\\ &<&\frac{1}{N}\sum_{n=1}^{n_0}a_n+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}\\ &=&\frac{1}{N}\sum_{n=1}^{n_0}a_n+\frac{2\varepsilon}{3}\,\cdot \end{array}

Taking {n_1} such that

\displaystyle \frac{1}{N}\sum_{n=1}^{n_0}a_n<\frac{\varepsilon}{3}\,,\ \ \forall\,n\ge n_1,

we get

\displaystyle \frac{1}{N}\sum_{n=1}^{N}a_n<\varepsilon\,,\ \ \forall\,n\ge\max\{n_0,n_1\},

which concludes the proof. \Box

Taking {a_n=\mu(T^{-n}A\cap B)} and {a=\mu(A)\cdot\mu(B)}, Lemma 4 implies that {\mathbb X} is weak mixing if and only if, for every {A,B\in\mathcal A}, there exists {E=E(A,B)\subset{\mathbb Z}} of zero density such that \displaystyle \lim_{n\rightarrow+\infty\atop{n\not\in E}}\mu(T^{-n}A\cap B)=\mu(A)\cdot\mu(B)\,. \ \ \ \ \ (4)

By approximation, (4) is equivalent to the existence, for every {f,g\in L^2(\mu)}, of a set {E=E(f,g)\subset{\mathbb Z}} of zero density such that \displaystyle \lim_{n\rightarrow+\infty\atop{n\not\in E}}\left(f\circ T^n,g\right)=(f,1)\cdot (1,g)\,. \ \ \ \ \ (5)

Another characterization comes from Proposition 3: {\mathbb X} is weak mixing if and only if \displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|\left(f\circ T^n,f\right)\right|^2=0\,, \ \ \ \ \ (6)

for every {f\in L^2(\mu)} such that {\int_X fd\mu=0}. In fact, by Lemma 4, {(a_n)} converges a Cesàro to zero if and only if the same happens to {({a_n}^2)}.

Weak mixing, at first impression, seems an artificial notion, obtained by the relaxation of strong mixing. This is not the case: first because it also has the natural spectral characterization discussed in section 1 and second, as was already discussed above, weak mixing is an important part of every mps. In other contexts, it is abundant. For example, Avila and Forni proved that almost every interval exchange transformation is either an irrational rotation or weak mixing, in contrast to an older result of Katok which proved that these are never strong mixing.

3. Product characterization of weak mixing

Consider two mps {\mathbb X=(X,\mathcal A,\mu,T)} and {\mathbb Y=(Y,\mathcal B,\nu,S)}.

Definition 5 The product mps of {\mathbb X} and {\mathbb Y} is the quadruple\displaystyle \mathbb X\times\mathbb Y=(X\times Y,\mathcal A\otimes\mathcal B,\mu\times\nu,T\times S),

where {\mathcal A\otimes\mathcal B} is the {\sigma}-algebra generated by {\mathcal A\times\mathcal B} and {\mu\times\nu} is the probability measure on {\mathcal A\otimes\mathcal B} defined by\displaystyle (\mu\times\nu)(A\times B)=\mu(A)\cdot\nu(B)\,,\ \forall\,A\in\mathcal A,B\in\mathcal B.

Theorem 6 Given a mps {\mathbb X=(X,\mathcal A,\mu,T)}, the following are equivalent.

  1. {\mathbb X} is weak mixing.
  2. {\mathbb X\times\mathbb X} is weak mixing.
  3. {\mathbb X\times\mathbb Y} is ergodic, for every ergodic mps {\mathbb Y}.
  4. {\mathbb X\times\mathbb X} is ergodic.

Proof: (i) {\Longrightarrow} (ii). It is enough to check (4) for a generating algebra of {\mathcal A\otimes\mathcal A}. Let {A_1,A_2,A_3,A_4\in\mathcal A}. By assumption, there exist {E_1,E_2\subseteq\mathbb Z} of zero density such that

\displaystyle \lim_{n\rightarrow\infty\atop{n\notin E_1}}\mu(T^{-n}A_1\cap A_3)=\mu(A_1)\cdot\mu(A_3)


\displaystyle \lim_{n\rightarrow\infty\atop{n\notin E_2}}\mu(T^{-n}A_2\cap A_4)=\mu(A_2)\cdot\mu(A_4).

The set {E=E_1\cup E_2} has zero density and satisfies

\displaystyle \begin{array}{rcl} &&\lim_{n\rightarrow\infty\atop{n\notin E}}(\mu\times\mu)\left((T\times T)^{-n}(A_1\times A_2)\cap (A_3\times A_4)\right)\\ &=& \lim_{n\rightarrow\infty\atop{n\notin E}}(\mu\times\mu)\left(T^{-n}A_1\cap A_3\times T^{-n}A_2\cap A_4\right)\\ &=&\lim_{n\rightarrow\infty\atop{n\notin E_1,E_2}}\mu(T^{-n}A_1\cap A_3)\cdot\mu(T^{-n}A_2\cap A_4)\\ &=&\mu(A_1)\mu(A_2)\mu(A_3)\mu(A_4)\\ &=&(\mu\times\mu)(A_1\times A_2)\cdot(\mu\times\mu)(A_3\times A_4)\,, \end{array}

proving that {T\times T} is weak mixing.

(ii) {\Longrightarrow} (i). Given {A,B\in\mathcal A}, there exists {E\subseteq{\mathbb Z}} of zero density such that

\displaystyle \begin{array}{rcl}&&\lim_{n\rightarrow\infty\atop{n\notin E}}(\mu\times\mu)((T\times T)^{-n}(A\times X)\cap(B\times X)) \\&=&(\mu\times\mu)(A\times X)\cdot(\mu\times\mu)(B\times X)\,,\end{array}

that is,

\displaystyle \lim_{n\rightarrow\infty\atop{n\notin E}}\mu(T^{-n}A\cap B)=\mu(A)\cdot\mu(B).

(i) {\Longrightarrow} (iii). Follows from the exercise below.

Exercise 3 Consider two bounded sequences {(a_n)_{n\ge 1}} and {(b_n)_{n\ge 1}} of real numbers. If {(a_n)} and {(b_n)} converge a Cesàro to {a} and {b}, respectively, then {(a_nb_n)} converges a Cesàro to {ab}.

(iii) {\Longrightarrow} (iv). If {\mathbb Y} is the trivial mps with {Y} consisting of a single point, we conclude that {\mathbb X} is ergodic. Then, takin {\mathbb Y=\mathbb X}, it follows that {\mathbb X\times\mathbb X} is ergodic.

(iv) {\Longrightarrow} (i). First, note that {\mathbb X} is ergodic. Given {A,B\in\mathcal A},

\displaystyle \mu((T\times T)^{-n}(A\times X)\cap(B\times X))=\mu(T^{-n}A\cap B)

converges a Cesàro to

\displaystyle (\mu\times\mu)(A\times X)\cdot(\mu\times\mu)(B\times X)=\mu(A)\cdot\mu(B)\,,

proving the assertion. Then

\displaystyle \begin{array}{rcl} &&\frac{1}{N}\sum_{n=1}^{N}|\mu(T^{-n}A\cap B)-\mu(A)\mu(B)|^2\\&=& \frac{1}{N}\sum_{n=1}^{N}\mu(T^{-n}A\cap B)^2\\ &&-2\mu(A)\mu(B)\cdot\frac{1}{N}\sum_{n=1}^{N}\mu(T^{-n}A\cap B)\\ &&+\mu(A)^2\cdot\mu(B)^2\\ &=&\frac{1}{N}\sum_{n=1}^{N}(\mu\times\mu)((T\times T)^{-n}(A\times A)\cap(B\times B))\\ &&-2\mu(A)\mu(B)\cdot\frac{1}{N}\sum_{n=1}^{N}\mu(T^{-n}A\cap B)\\ &&+\mu(A)^2\mu(B)^2, \end{array}

which converges to

\displaystyle \mu(A)^2\mu(B)^2-2\mu(A)^2\mu(B)^2+\mu(A)^2\mu(B)^2=0\,.


4. Spectral characterization of weak mixing

We now characterize weak mixing in terms of spectral properties. At this point, it is interesting to introduce the

Theorem 7 If {U} is an unitary operator on the Hilbert space {(\mathcal H,(\,,\,))} and {x\in\mathcal H}, then there is a unique finite Borel measure {\nu_x} on the circle {\mathbb S^1} such that\displaystyle (U^nx,x)=\int_{\mathbb S^1}z^nd\nu_x\ ,\ \ n=0,1,\ldots

When {T} has continuous spectrum, {\nu_f} is a continuous measure (it has no atoms), for every {f\in L^2(\mu)} such that {\int fd\mu=0}.In this case, Fubini theorem guarantees that {\nu_f\times\nu_f} gives zero measure to the diagonal {\Delta=\{(z,z)\,,\,z\in\mathbb S^1\}}. This in turn implies the

Theorem 8 {\mathbb X} is weak mixing if and only if {T} has continuous spectrum.

Proof: ({\Longrightarrow}) Suppose {f\not=0} is an eigenfunction associated to {\lambda\not=1}. The function {g:X\times X\rightarrow{\mathbb R}} defined by {g(x_1,x_2)=f(x_1)\cdot\overline f(x_2)} is an eigenfunction of {U_{T\times T}} associated to {1}. By Theorem 6, {g} is constant and the same happens to {f}.

({\Longleftarrow}) Let us check (6). Take {f\in L^2(\mu)} such that {\int_X fd\mu=0}. Using Theorem 7,

\displaystyle \begin{array}{rcl} \dfrac{1}{N}\sum_{n=1}^N\left|\int_X(f\circ T^n)\overline fd\mu\right|^2 &=&\dfrac{1}{N}\sum_{n=1}^N\left|\int_{\mathbb S^1}z^nd\nu_f(z)\right|^2\\ &=&\dfrac{1}{N}\sum_{n=1}^N\int_{\mathbb S^1}z^nd\nu_f(z)\cdot\int_{\mathbb S^1}\overline w^nd\nu_f(w)\\ &=&\dfrac{1}{N}\sum_{n=1}^N\int_{\mathbb S^1\times\mathbb S^1}(z\overline w)^nd(\nu_f\times\nu_f)(z,w)\\ &=&\int_{\mathbb S^1\times\mathbb S^1}\dfrac{1}{N}\sum_{n=1}^N(z\overline w)^nd(\nu_f\times\nu_f)(z,w)\,. \end{array}

Decompose {\mathbb S^1\times\mathbb S^1=\Delta\cup\Sigma}, where {\Delta=\{(z,z)\,,\,z\in\mathbb S^1\}} is the diagonal. For {(z,w)\in\Sigma}, the summand

\displaystyle \dfrac{1}{N}\sum_{n=1}^N(z\overline w)^n=\dfrac{(z\overline w)^{N+1}-1}{N\cdot(z\overline w-1)}

converges to zero as {N\rightarrow+\infty} uniformly in {\Sigma}. Since {\nu_f\times\nu_f} assigns zero measure to {\Delta}, we’re done. \Box

5. Conditions for weak mixing

In this section we resume all conditions obtained above for a a mps {\mathbb X=(X,\mathcal A,\mu,T)} be weak mixing.

  1. For any {A,B\in\mathcal A},\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|\mu(T^{-n}A\cap B)-\mu(A)\cdot\mu(B)\right|=0\,.
  2. For any {f,g\in L^2(\mu)},\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|(f\circ T^n,g)-(f,1)\cdot(1,g)\right|=0\,.
  3. For any {f\in L^2(\mu)} such that {\int_Xfd\mu=0},\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|(f\circ T^n,f)\right|=0\,.
  4. For any {f\in L^2(\mu)} such that {\int_Xfd\mu=0},\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|(f\circ T^n,f)\right|^2=0\,.
  5. For any {A,B\in\mathcal A}, there exists {E\subset{\mathbb Z}} of zero density such that\displaystyle \lim_{n\rightarrow+\infty\atop{n\not\in E}}\mu(T^{-n}A\cap B)=\mu(A)\cdot\mu(B)\,.
  6. For any {f,g\in L^2(\mu)}, there exists {E\subset{\mathbb Z}} of zero density such that\displaystyle \lim_{n\rightarrow+\infty\atop{n\not\in E}}(f\circ T^n,g)=(f,1)\cdot (1,g)\,.
  7. {\mathbb X\times\mathbb X} is ergodic.
  8. {\mathbb X\times\mathbb Y} is ergodic, for every {\mathbb Y} ergodic.
  9. {\mathbb X\times\mathbb X} is weak-mixing.
  10. {T} has continuous spectrum.

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6, ERT7.


  1. I have a question, in the proof of theorem 8, ¿Why is necessary to (f,1)=0?
    If (f,1) is not 0, can we say that T is Weak mixing?

    Sorry for my english

    • The notion of weak-mixing can be defined in terms of the behavior of general L^2 observables (cf. Exercise 2) or just L^2 observables f with zero mean 0=(f,1) (cf. Proposition 3) because any observable f is naturally associated to the observable g = f- (f,1) with zero mean.

      In fact, the sole difference between these definitions is that the equation (3) is more compact than (2). In particular, one prefers to use (3) rather than (2) when proving statements about weak mixing just for the sake of simplicity.

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