Special Case Where Edges Transform Continuously or in Measure Preserving Way Over Time

Measure Preserving Transformation

A generic measure preserving transformation in the weak topology is weakly mixing (hence ergodic), rigid (hence is not mildly mixing), has simple singular spectrum such that the maximal spectral type in L02 together with all its convolutions are mutually singular and supported by a thin set on any given scale.

From: Handbook of Dynamical Systems , 2006

Handbook of Dynamical Systems

Anatole Katok , Jean-Paul Thouvenot , in Handbook of Dynamical Systems, 2006

5.8.3 Homogeneous spectrum of arbitrary multiplicity and group actions

Measure preserving transformations with homogeneous spectrum of arbitrary multiplicity (including new examples with multiplicity two) were recently found by Ageev [17] using a different type of symmetry. His main idea is quite brilliant although in retrospect it looks natural.

Ageev considers the following group G_m . It is a finite extension of ℤ ^m and has generators T ₁, …, T_m , S where T ₁, …, T_m commute, T ₁·T ₂·····T_m = Id and T_{i +1} = S ··T_i · S ⁻¹ for i = 1, …, m −1. Notice that S^m commutes with T ₁, …, T_m and thus the group G_m is an m-fold extension of the Abelian group with generators T ₁, …, T_{m −1}, S^m .

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Functional Analysis and its Applications

Oksana A. Ochakovskaya , in North-Holland Mathematics Studies, 2004

1 Introduction

Measure-preserving transformations plays an important role in ergodic theory and related questions (see, for instance, [1], [2] and the bibliography therein). In the present paper we study such transformations acting from ℝ ⁿ into ℝ ⁿ . Throughout, we assume that n ≥ 2.

Denote by C^k (ℝ ⁿ , ℝ ⁿ ) the collection of all C^k – mappings from ℝ ⁿ into ℝ ⁿ . For r > 0, y ∈ ℝ ⁿ we set B_r (y)   =   {x ∈ ℝ ⁿ : |x – y| ≤ r}, S_r (y)   =   {x ∈ ℝ ⁿ : |x – y|   = r}, where | · | is the Euclidean norm in ℝ ⁿ . We write m(G) for Lebesgue measure of the set G ⊂ ℝ ⁿ . Let f *g be a convolution of functions f and g. Denote by χ_G the characteristic function of the set G ⊂ ℝ ⁿ . For r > 0 let V_r (ℝ ⁿ ) be a collection of all functions f ∈ C(ℝ ⁿ ) with zero integrals over each ball with radius r. We set also Λ _n   =   { $t>0:t=\frac{\lambda }{\beta }$ , where $J_{\frac{n}{2}}\left(\alpha \right)=J_{\frac{n}{2}}\left(\beta \right)=0,\beta \ne 0$ }, where J_v is the Bessel function of order v. Let Eⁿ   =   {x ∈ Rⁿ : x_n > 0}.

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Handbook of Dynamical Systems

Vitaly Bergelson , ... Máté Wierdl , in Handbook of Dynamical Systems, 2006

Theorem 4.21

For any commuting measure preserving transformations T ₁ , T ₂ , …, T_k of a probability space (X, B, μ)and for any A ∈ B with μ (A) > 0 one has

$\underset{N\to \infty }{\lim \inf }\frac{1}{N}{\displaystyle \sum _{n-0}^{N-1}\mu \left(A\cap T_1^{-n}A\cap \cdots \cap T_k^{-n}A\right)>0.}$

The main new difficulty which one faces when dealing with k general commuting transformations is that they generate a ℤ^k-action, which may have different dynamical properties along the sub-actions of different subgroups. In other words, while Theorem 4.2 was about the joint behavior of k commuting transformations of a special form, namely T, T ² , …, T^k , in Theorem 4.21 we have to study k commuting transformations which are in, so to say, general position. This complicates the underlying structure theory, which has to be "tuned up" to reflect the more complicated situation when different operators in the group generated by T ₁ , …, T_k have different dynamical properties. What saves the day is Theorem 4.24 below, which is at the core of Furstenberg and Katznelson's proof of Theorem 4.21.

We need first to introduce some pertinent definitions. While these definitions make sense for any measure preserving group actions (and are given below a general formulation for future reference), the reader should remember that in the discussion of the proof of Theorem 4.19, the group G which occurs in the next two definitions is meant to stand for ℤ ^k (and hence the subgroups of G are themselves isomorphic to ℤ ^l for some 0 ≤ l ≤ k).

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Handbook of Dynamical Systems

Amos Nevo , in Handbook of Dynamical Systems, 2006

Remark 10.10 (Generalization of Birkhoff's theorem)

Given an arbitrary invertible measure preserving transformation T on a probability space X, Birkhoff's pointwise ergodic theorem asserts that for any f ∈ L ¹(X), the averages of f along an orbit of T, namely the expressions $\frac{f\left(T^{-n}x\right)+\cdots +f\left(T^{n}x\right)}{2n+1}$ converge, for almost all x ∈ X, to the limit $\tilde{f}\left(x\right)$ , where $\tilde{f}$ is the conditional expectation of f w.r.t. the σ-algebra of T-invariant sets. Part of our quest to establish ergodic theorems for group actions can thus be motivated by the following obvious and natural question which presents itself. Given two arbitrary invertible measure preserving transformations T and S, find a geometrically natural way to average a function f along the orbits of the group generated by T and S, so as to obtain the same conclusion.

Of course if T and S happen to commute, then, according to the discussion in Section 5, the expressions $\frac{1}{{\left(2n+1\right)}^2}{\displaystyle {\sum }_{-n\le n_1,n_2\le n}f\left(T^{n_1}{S}^{n_2}x\right)}$ converge for almost all x ∈ X, for any f ∈ L ¹(X), and again the limit is the conditional expectation of f w.r.t. the σ-algebra of sets invariant under T and S. In other words, the pointwise ergodic theorem holds for finite-measure-preserving actions of the free Abelian group on two generators, namely ℤ². However, it is clear that when choosing generically two volume-preserving diffeomorphisms of a compact manifold, or two orthogonal transformations of the Euclidean unit sphere, or in general two measure preserving maps of a given measure space, the group generated by them is not Abelian, and in fact, it is generically free.

The answer to the problem above is then to find an averaging sequence satisfying a pointwise ergodic theorem for finite-measure-preserving actions of the free non-Abelian group on two generators. The first choice that one would consider by direct analogy with Birkhoff's and Wiener's theorems (for ℤ and ℤ^d), would be the normalized ball averages w.r.t. a set of free generators. For the free group this problem has been settled, using the spectral methods described above, by the following result.

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Handbook of Dynamical Systems

Luis Barreira , ... Omri Sarig , in Handbook of Dynamical Systems, 2006

5.7 The case of noninvertible dynamical systems

Consider a measure preserving transformation f : X → X of a Lebesgue space (X, ν) (the map f need not be invertible). We assume that ν is a probability measure. Given a measurable function $A:X\to GL\left(n,ℝ\right)$ and x ∈ X, define the one-sided cocycle $\mathcal{A}:X\times \mathcal{N}\to GL\left(n,ℝ\right)$ by

$\mathcal{A}\left(x,m\right)=A\left(f^{m-1}\left(x\right)\right)\dots A\left(f\left(x\right)\right)A\left(x\right).$

Note that the cocycle equation (4.1) holds for every $m,k\in \mathcal{N}.$ Given $\left(x,v\right)\in X\times {ℝ}^n,$ define the forward Lyapunov exponent of (x, v) (with respect to $\mathcal{A}$ ) by

${\chi }^{+}\left(x,v\right)={\chi }^{+}\left(x,v,\mathcal{A}\right)=\underset{m\to +\infty }{\overline{\lim }}\frac{1}{m}\log \Vert \mathcal{A}\left(x,t\right)v\Vert .$

However, since the map f and the matrices A (x) may not be invertible, one may not in general define a backward Lyapunov exponent. Therefore, we can only discuss the forward regularity for $\mathcal{A}$ . One can establish a Multiplicative Ergodic Theorem in this case.

Theorem 5.17

Let f be a measure preserving transformation of a Lebesgue space (X, ν), and $\mathcal{A}$ a measurable cocycle over f such that log⁺ || A || ∈ L ¹(X, ν). Then the set of forward regular points for $\mathcal{A}$ has full ν-measure and for ν-almost every x ∈ X and every subspace $F\subset E_i^{+}\left(x\right)$ such that $F\cap E_{i-1}^{+}\left(x\right)=\left\{0\right\}$ we have

$\underset{m\to +\infty }{\lim }\frac{1}{m}\log \underset{v}{\inf }\Vert \mathcal{A}\left(x,m\right)v\Vert =\underset{m\to +\infty }{\lim }\frac{1}{m}\log \underset{v}{\sup }\Vert \mathcal{A}\left(x,m\right)v\Vert ={\chi }_i^{+}\left(x\right),$

with the infimum and supremum taken over {v ∈ F: || v || = 1}.

When the matrix A (x) is invertible for every x ∈ X and log⁺ || A ||, log⁺ || A ⁻¹ || ∈ L ¹(X, ν) for some f-invariant Lebesgue measure ν, one can show that for the cocycle induced by $\mathcal{A}$ on the inverse limit of f the set of regular points has full ν-measure.

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Handbook of Dynamical Systems

Yuri Kifer , Pei-Dong Liu , in Handbook of Dynamical Systems, 2006

Jacobian

One ingredient is to overcome the difficulty caused by the fact that the random maps here are in general not one-to-one but the usual inverse limit space method does not seem helpful. The notion of the Jacobian of a measure-preserving transformation introduced by [132] turns out to be very useful for dealing with local homeomorphisms. The definition is as follows.

Let f : X → Y be a measure-preserving transformation between two probability spaces $\left(X,\text{A},\nu \right)$ and (Y, ℬ, ρ). Assume that there is a countable measurable partition α = {A_i } of X (ν-mod 0) such that for each A_i the map f_i := f : | _Ai : A_i → Y is absolutely continuous, that is,

(i)

f_i is injective;

(ii)

f_i (A) is measurable if A is a measurable subset of A_i ;

(iii)

ρ(f_i (A)) = 0 if A ⊂ A_i is measurable and ν(A) = 0.

By (i) and (ii) we can define a measure ${\nu }_{f_i}$ on each A_i by ${\nu }_{f_i}\left(A\right)=\rho \left(f_i\left(A\right)\right)$ for measurable set A ⊂ A_i . By (iii), ${\nu }_{f_i}$ is absolutely continuous with respect to ${\nu }_i:={\nu |}_{A_i}$ . Define a measurable function $J\left(f\right):X\to {ℝ}^{+}$ by

$\begin{array}{ll}J\left(f\right)\left(x\right)=\frac{d{\nu }_{f_i}}{d{\nu }_i}\left(x\right)\hfill & \text{if}x\in A_i.\hfill \end{array}$

It is easy to see that the definition of J (f) is independent of the choice of partition α, and we will call J (f) the Jacobian of f. Clearly, J (f)(x) ≥ 1 for μ-a.e. x ∈ X. As an exercise, consider the following simple situation: if f : M → M is a C ¹ map with no singularities and ν is a Borel probability on M such that ν ≪ Leb, then f: (M, ν) → (M, f ν) admits a Jacobian J (f) given by

(3.1.15) $\begin{array}{ll}J\left(f\right)\left(x\right)=\frac{\left({ℒ}_{f}l\right)\left(fx\right)}{l\left(x\right)}\left|\det D_{x}f\right|,\hfill & \nu \text{-a}.\text{e}\text{.}x\in M,\hfill \end{array}$

where l is the density of ν with respect to the Lebesgue and ℒ_fl is defined by

(3.1.16) $\left({ℒ}_{f}l\right)\left(x\right)={\displaystyle \sum _{y:fy=x}\frac{l\left(y\right)}{\left|\det D_{y}f\right|}}$

(note that ${ℒ}_{f}l=l$ ν-a.e. is equivalent to f ν = ν).

We now state a very useful property of this notion. Assume moreover that $\left(X,\text{A},\nu \right)$ and (Y, ℬ, ρ) are both Lebesgue spaces. If ξ is a measurable partition of Y, by ${\left\{{\nu }_x^{f^{-1}\xi }\right\}}_{x\in X}$ and ${\left\{{\rho }_y^{\xi }\right\}}_{y\in Y}$ we will denote, respectively, a canonical system of conditional measures of ν and ρ associated with $f^{-1}\xi$ and ξ.

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Ergodic Theory

M. Yuri , in Encyclopedia of Mathematical Physics, 2006

Convergence to Equilibrium States and Mixing Properties

Let T  : X → X be a measure-preserving transformation on a probability space $\left(X,\mathcal{B},\mu \right)$ . We call T to be "weak mixing" if for any $A,B\in \mathcal{B}$

$\underset{N\to \infty }{lim}\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}}|\mu (T^{-n}A\cap B)-\mu (A)\mu (B)|=0$

The weak-mixing property of (T, μ) can be represented by; $\forall f,g\in L^2\left(X,\mathcal{B},\mu \right)$

$\underset{N\to \infty }{lim}\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}}|{\int }_X(f{T}^n)g\text{d}\mu -{\int }_{X}f\text{d}\mu {\int }_{X}g\text{d}\mu |=0$

and this is equivalent to the ergodicity of (T × T, μ × μ). Moreover, (T, μ) is weak mixing if and only if the unitary operator $U:\mathcal{H}\to \mathcal{H}$ defined by Uf(x) = f(Tx) has no eigenfunctions that are not constants (μ  mod   0). We say that the operator U has continuous spectrum if there are no eigenvectors. If $\mathcal{H}$ is the closure of the linear span of the eigenvectors, then we say that the operator U has pure point spectrum. The weak-mixing property of (T, μ) just implies that U restricted on the orthonormal subspace of the subspace consisting of constant functions has continuous spectrum. We recall that if U has one as a simple eigenvalue then T is ergodic. Additionally, if there are no other eigenvalues, then T is weakly mixing. Hence, if T is weak mixing, then it is necessarily ergodic. The next property corresponds to the term "relaxation" in physics literature which is used to describe processes under which the system passes to a certain stationary state independently of its original state. We call T (strong) mixing if for any $A,B\in \mathcal{B}$

$\underset{n\to \infty }{lim}\mu (T^{-n}A\cap B)=\mu (A)\mu (B)$

Then (T, μ) is (strong) mixing if and only if for any $f,g\in L^2(X,\mathcal{B},\mu )$

$\underset{n\to \infty }{lim}{\int }_X(f{T}^n)g\text{d}\mu ={\int }_{X}f\text{d}\mu {\int }_{X}g\text{d}\mu$

and mixing is necessarily weak mixing. Moreover, for any probability measure ν absolutely continuous with respect to μ, one can show that lim _n→∞ ν(T ⁻ⁿ A) = μ(A) for every $A\in \mathcal{B}$ . Thus, any nonequilibrium distribution tends to an equilibrium one with time. The mixing property has a significant meaning from a physical point of view, as it implies decay of correlation of observable functions; moreover, limiting distributions of averaged observables are determined by the decay rates of correlation functions for many cases (e.g., hyperbolic systems). For any $f\in L^2(X,\mathcal{B},\mu )$ we consider the scalar products s _n = s _n (f) = (U ⁿ f, f), n ≥ 0 and define $s_n:={\bar{s}}_{-n}$ for n < 0. The sequence {s _n } _n∈Z is positive definite and so by Bohner's theorem, we can write $s_n\left(f\right)={\int }_0^{1}exp\left[2\pi \text{i}n\lambda \right]\text{d}{\sigma }_f\left(\lambda \right)$ , where σ _f is a finite Borel measure on the unit circle S ¹ and satisfies the condition that σ _f (S ¹) = ||f||². Such a measure is called a spectral measure of f. We see that T is mixing iff for any $f\in L^2(X,\mathcal{B},\mu )$ with ∫ _X f  dμ = 0 the Fourier coefficients {s _n } of the spectral measure σ _f tend to zero as |n| → ∞. Let $\left(X,\mathcal{B},\mu \right)$ be isomorphic to $\left(\left[0,1\right],{\mathcal{B}}_0,\lambda \right)$ , where ${\mathcal{B}}_0$ is the Borel σ-algebra on [0, 1] and λ is the normalized Lebesgue measure of [0, 1]. Then we call a measure-preserving transformation T on $\left(X,\mathcal{B},\mu \right)$ an exact endomorphism if ${\cap }_{n=0}^{\infty }{T}^{-n}\mathcal{B}=\{X,\varnothing \}(\mu mod0)$ . We can verify that an exact endomorphism is (strong) mixing (Rohlin 1964). Moreover, μ is exact if for any positive-measure set $A\in \mathcal{B}$ with $T^{n}A\in \mathcal{B}(\forall n\ge 0){lim}_{n\to \infty }\mu \left(T^{n}A\right)=1$ holds. Let T be a nonsingular transformation on $(X,\mathcal{B},\nu )$ , that is, νT ⁻¹ ∼ ν. Then we can define the transfer (Perron–Frobenius) operator ${\mathcal{L}}_{\nu }:L^1(X,\nu )\to L^1(X,\nu )$ by ${\mathcal{L}}_{\nu }f:=\text{d}\left(f\nu \right)T^{-1}/\text{d}\nu$ , which satisfies

${\int }_X({\mathcal{L}}_{\nu }f)g\text{d}\nu ={\int }_{X}f(gT)\text{d}\nu (\forall g\in L^{\infty }(X,\nu ))$

We say that a nonsingular measure ν is exact if $A\in {\cap }_{n=0}^{\infty }{T}^{-n}\mathcal{B}$ implies ν(A)ν(A ^c ) = 0. By Lin's theorem (Lin 1971) the exactness of ν can be described as follows; $\forall f\in L^1\left(X,\nu \right)$ with ${\int }_{X}f\text{d}\nu =0,{lim}_{n\to \infty }\left|\right|{\mathcal{L}}_{\nu }^{n}f|{|}_1=0$ . Let μ = hν be an exact T-invariant probability measure equivalent to ν. Then the upper bounds of mixing rates of the exact measure μ = hν are determined by the speed of L ¹-convergence of the iterated transfer operators $\left\{{\mathcal{L}}_{\nu }^n\right\}$ . This is because ${\mathcal{L}}_{\nu }h=h$ and for every f ∈ L ¹(X, ν) with ${\int }_{X}f\text{d}\nu =1,{lim}_{n\to \infty }\left|\right|{\mathcal{L}}_{\nu }^{n}f-h|{|}_1=0$ . Hence, the property ${\mathcal{L}}_{\mu }f=h^{-1}{\mathcal{L}}_{\nu }(hf)$ allows one to see that for every f, g ∈ L ^∞(X, μ) the correlation function

$C_{f,g}(n):=\left|{\int }_X(f{T}^n)g\text{d}\mu -{\int }_{X}f\text{d}\mu {\int }_{X}g\text{d}\mu \right|$

is bounded from above by

$\begin{array}{l}\left|\right|f\left|{|}_{\infty }\right||{\mathcal{L}}_{\mu }^{n}g-{\int }_{X}g\text{d}\mu |{|}_1\\ =\left|\right|f\left|{|}_{\infty }\right|\left|h^{-1}\right\{{\mathcal{L}}_{\nu }^n(gh)-P(gh)\left\}\right|{|}_1\end{array}$

where P  : L ¹(X, ν) → L ¹(X, ν) is a linear operator defined by Pf: = h ∫ _X f  dν. The operator P is the one-dimensional projection operator associated to the eigenvalue 1 (which is maximal in many cases) of ${\mathcal{L}}_{\nu }$ satisfying P ² = P and $P{\mathcal{L}}_{\nu }={\mathcal{L}}_{\nu }P=P$ . Moreover, since ${\mathcal{L}}_{\nu }^n-P={({\mathcal{L}}_{\nu }-P)}^n$ , the exponential decay of mixing rates follows from the spectral gap of ${\mathcal{L}}_{\nu }$ , that is, 1 is the simple isolated maximal eigenvalue of ${\mathcal{L}}_{\nu }$ .

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Ergodic Theory

Frank Blume , in Handbook of Measure Theory, 2002

THEOREM 8.1

(Arnoux et al., 1985). Any aperiodic measure-preserving transformation on a probability space is isomorphic to an interval exchange transformation T on [0, 1 [, that is a map T:[0,1[→[0,1[which satisfies the following properties:

(a)

There exists a strictly increasing sequence ${\left\{t_n\right\}}_{n\in \mathbb{N}}\subset [0,1[andasequence\left\{a_n\right\}\subset ℝ$ such that t_o = 0, lim_n∞ = 1 and T(x) = x + a_n for all x ∈:=[t_n-1,t_n[.

(b)

$T\left(I_n\right)\subset [0,1[$ for all n ∈ N

(c)

T is bijective.

Adic transformations. An interesting combinatorial approach for representing cutting-and-stacking procedures (that is aperiodic measure-preserving transformations) has been introduced by Vershik (1989, 1991). The so-called adic transformations are defined on the set X of all infinite paths in a graph which has the following characteristics:

(a)

The graph consists of infinitely many levels with a finite number of vertices in each level.

(b)

The vertices in each level are connected only to vertices in the levels directly above and below.

(c)

In each level the vertices are ordered from left to right.

A σ-algebra on X is naturally generated by the cylinder sets of infinite paths. Furthermore, we introduce a partial order on X such that two paths

$\begin{array}{lll}x=\left(x_1,x_2,x_3,...\right)\hfill & \text{and}\hfill & y=\left(y_1,y_2,y_3,...\right)\hfill \end{array}$

are comparable if there is an n ∈ N such that x_k=y_k for all k > n. For any pair of comparable paths x and y we say that x < y if x_n < y_n for the largest integer n for which x_n ≠ y_n . For a given nonmaximal path x we define Tx to the smallest path y which is larger than x. In order for T to represent a cutting and stacking process, it is necessary to remove from X a countable set of minimal and maximal elements (for details see also Vershik and Livshits (1992)). To illustrate how a cutting-and-stacking procedure can be represented by an adic transformation, we consider an example:

Cutting-and-stacking procedure:

Equivalent adic transformation:

The measure of a cylinder set $C=\left\{x\in X|x_1=a_1,...,x_n=a_n\right\}$ is here $\frac{1}{2^n}$ .

Furthermore, the 2ⁿ cylinder sets of the form $\left\{x\in X|x_1=a_1,...,x_n=a_n\right\}$ represent the 2ⁿ levels obtained in the nth step of the cutting-and-stacking procedure.

Rokhlin towers. The usefulness of the so-called Rokhlin tower decomposition for the efficient formulation of proofs and as a basis for various constructions in ergodic theory depends on the following theoremml:

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Handbook of Dynamical Systems

E. Glasner , B. Weiss , in Handbook of Dynamical Systems, 2006

Corollary 11.2

Let (X, χ, μ, T) be an ergodic measure preserving transformation with infinite point spectrum defined by (G, ρ) where G is a compact monothetic group $G={\overline{\left\{{\rho }^n\right\}}}_{n\in \mathbb{Z}}$ . Then there is an almost 1-1 minimal extension of (G, ρ) (i.e. a minimal almost automorphic system), $\left(\tilde{Z},\sigma \right)$ and an invariant measure ν on Z such that (Z, σ, ν) is isomorphic to (X, χ, μ, T).

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Probability on MV-Algebras

Beloslav Rieĕan , Daniele Mundici , in Handbook of Measure Theory, 2002

THEOREM 4.16

Let (Ω, S, P) be a probability space equipped with a measure preserving transformation T. Let F be the full tribe of all functions f : Ω → [0, 1] which are measurable with respect to S. Let U : F → F be given by U(f) = f o T. Suppose the boolean partition $A^{*}=\left(g_l^{*},\dots ,g_k^{*}\right)$ is a generator of S. Also assume G to be a subset of F such that $g_l^{*}\in G$ for each i = 1,…,k. Then $h_{U.G}^b=h^b\left(A^{*},U\right)=h\left(A^{*},U\right).$

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