Measure-theoretic mean equicontinuity and bounded complexity

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ScienceDirect J. Differential Equations 267 (2019) 6152–6170 www.elsevier.com/locate/jde

Measure-theoretic mean equicontinuity and bounded complexity Tao Yu Shanghai center for mathematical sciences, Fudan University, Shanghai, 200438, PR China Received 4 April 2019; accepted 20 June 2019 Available online 28 June 2019

Abstract Let (X, B, μ, T ) be a measure preserving system. We say that a function f ∈ L2 (X, μ) is μ-mean  k  equicontinuous if for any  > 0 there is k ∈ N and measurable sets A1 , A2 , · · · , Ak with μ Ai > 1 −  such that whenever x, y ∈ Ai for some 1 ≤ i ≤ k, one has

i=1

n−1 1 |f (T j x) − f (T j y)| < . n→∞ n j =0

lim sup

Measure-theoretic complexity with respect to f is also introduced. It is shown that f is an almost periodic function if and only if f is μ-mean equicontinuous if and only if μ has bounded complexity with respect to f . Ferenczi studied measure-theoretic complexity using α-names of a partition and the Hamming distance. He proved that if a measure preserving system is ergodic, then the complexity function is bounded if and only if the system has discrete spectrum. We show that this result holds without the assumption of ergodicity. © 2019 Elsevier Inc. All rights reserved. MSC: 37A35 Keywords: Mean equicontinuity; Bounded complexity; Almost periodic function; Discrete spectrum

E-mail address: [email protected]. https://doi.org/10.1016/j.jde.2019.06.017 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction Throughout this paper, a topological dynamical system (t.d.s. for short) is a pair (X, T ), where X is a compact metric space with a metric d and T is a homeomorphism from X to itself, and a measure preserving system (m.p.s. for short) is a quadruple (X, B, μ, T ), where (X, B, μ) is a Lebesgue space and T is an invertible measure-preserving transformation from X to itself. There is a natural connection between topological dynamical systems and measure preserving systems. If (X, T ) is a t.d.s., then a probability measure μ on (X, BX ) is T -invariant if and only if (X, BX , μ, T ) is a m.p.s., where BX is the Borel σ -algebra of X. A t.d.s. (X, T ) is called equicontinuous if for any ε > 0 there exists δ > 0 such that d(T n x, T n y) < ε for any n ∈ N whenever d(x, y) < δ. The analogous concept of equicontinuity for m.p.s. is the discrete spectrum property. In the study of discrete spectrum, Fomin introduced the concept of stability in the mean in the sense of Lyapunov (mean-L-stability for short) in [3]. Li, Tu and Ye [11] proved that mean-L-stability is equivalent to mean equicontinuity, every ergodic measure in a mean equicontinuous system has discrete spectrum. In [8], Huang, Lu and Ye studied measure-theoretic equicontinuity for invariant measures of t.d.s. They showed that for an invariant measure μ of a t.d.s. (X, T ), if (X, T , μ) is μ-equicontinuous then μ has discrete spectrum. Following this idea, in [5] García-Ramos introduced measure-theoretic mean equicontinuity for an ergodic invariant measure μ of a t.d.s. (X, T ) and showed that (X, T , μ) is μ-mean equicontinuous if and only if μ has discrete spectrum, see also [10] for another proof. It should be noticed that recently the authors in [7,9] showed that for an invariant measure μ of a t.d.s. (X, T ), (X, T , μ) is μ-mean equicontinuous if and only if μ has discrete spectrum. Complexity function is very useful to describe equicontinuity. In [1], Blanchard et al. studied topological complexity via the complexity function of an open cover and showed that the complexity function is bounded for any open cover if and only if the system is equicontinuous. In [7, 9], Huang et al. studied topological and measure-theoretic complexity via a sequence of metrics n−1  d(T i x, T i y), and showed that an invariant measure μ on (X, T ) has d¯n where d¯n (x, y) = n1 i=0

bounded complexity with respect to d¯n if and only if (X, T ) is μ-mean equicontinuous if and only if μ has discrete spectrum, see also [13] for another proof. Ferenczi [2] studied measure-theoretic complexity of a m.p.s. (X, B, μ, T ) using α-names of a partition and the Hamming distance. He proved that if a m.p.s. is ergodic then the complexity function is bounded for any partition if and only if the system has discrete spectrum. In this paper, we will show that for an invariant measure μ on (X, T ), (X, T ) is μ-mean equicontinuous if and only if the complexity function of (X, BX , μ, T ) using α-names of a partition and the Hamming distance is bounded. Huang et al. [7] proved that (X, T ) is μ-mean equicontinuous if and only if it has discrete spectrum, then we know Ferenczi’s result also holds for invariant measure. Following the idea in [12], García-Ramos and Marcus [6] introduced measure-theoretic mean equicontinuity with respect to a function. Let (X, T ) be a t.d.s. with metric d and μ be an invariant measure on X, a function f ∈ L1 (X, μ) is (μ, d)-mean equicontinuous if for every τ > 0 there exists a compact subset K of X with μ(K) > 1 − τ such that for any ε > 0 there exists δ > 0 such that 1 |f (T i x) − f (T i y)| < ε n n−1

lim sup n→∞

i=0

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whenever d(x, y) < δ with x, y ∈ K. It was shown in [6] that for an ergodic measure μ on (X, T ), f ∈ L2 (X, μ) is (μ, d)-mean equicontinuous if and only if f is an almost periodic function, that is {f ◦ T n : n ∈ Z} is precompact in L2 (X, μ). In this paper, we use complexity function to describe measure-theoretic mean equicontinuity with respect to a function. For a function f ∈ L1 (X, μ), we define a sequence of pseudo-metrics n−1  |f (T i x) − f (T i y)|. We consider measure-theoretic complexity with respect f¯n (x, y) = n1 i=0

to {f¯n } and prove for an invariant measure μ on (X, T ) and f ∈ L2 (X, μ), μ has bounded complexity with respect to {f¯n } if and only if f is (μ, d)-mean equicontinuous if and only if f is an almost periodic function. We show that the results above also hold for a m.p.s. (X, B, μ, T ). The only difficulty to overcome is that the definition of μ-mean equicontinuity of a function relies on a given metric on X, even though it does not depend on the metric. We will give a purely measure-theoretical definition about μ-mean equicontinuity of a function which is equal to the ordinary one in t.d.s., which answers the question in [6]. Let (X, B, μ, T ) be a m.p.s., we say that a function f ∈ L1 (X, μ) is μ-mean if for any  > 0 there is k ∈ N and measurable sets A1 , A2 , · · · , Ak with  k equicontinuous   Ai > 1 −  such that whenever x, y ∈ Ai for some 1 ≤ i ≤ k, one has μ i=1

1 |f (T j x) − f (T j y)| < . n n−1

lim sup n→∞

j =0

The structure of the paper is the following. In Section 2, we recall some basic notions which we will use in the paper. In Section 3, we show that for a m.p.s. (X, B, μ, T ), complexity function using α-names of a partition and the Hamming distance is bounded if and only if the system has discrete spectrum. In Section 4, we give a purely measure-theoretical definition of μ-mean equicontinuity of a function, and show that it is equal to the ordinary one in t.d.s. We will also prove for a t.d.s. (X, T ) with an invariant measure μ, a function f ∈ L1 (X, μ) is μ-mean equicontinuous if and only if μ has bounded complexity with respect to {f¯n }. In Section 5, we show that f ∈ L2 (X, μ) is μ-mean equicontinuous if and only if f is an almost periodic function. Acknowledgments. I would like to thank Professors Xiangdong Ye, Jian Li and Guohua Zhang for useful suggestions. I was supported by China Postdoctoral Science Foundation, grant no. 2018M631996. 2. Preliminaries In this section we recall some notions and aspects of dynamical systems. Let A be a finite set, denote by #(A) the number of elements of A. 2.1. Measure-theoretic mean equicontinuity Let (X, T ) be a t.d.s. with a metric d. Denote the collection of all T -invariant Borel probability measures on (X, T ) by M(X, T ), and all ergodic measures of M(X, T ) by M e (X, T ). We say that a compact subset K of X is a mean equicontinuous set if for every  > 0, there exists a δ > 0 such that for all x, y ∈ K with d(x, y) < δ we have

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1 d(T i x, T i y) < . n n−1

lim sup n→∞

i=0

Let μ ∈ M(X, T ). We say that (X, T ) is μ-mean equicontinuous if for every τ > 0 there exists a mean equicontinuous set K of X with μ(K) > 1 − τ . 2.2. Lebesgue space A probability space (X, B, μ) is a Lebesgue space if it is isomorphic to a probability space which is the disjoint union of a countable (or finite) number of points {y1, y2 , · · · } each of positive measure and the space ([0, s], L[0, s], l) where L[0, s] is the σ -algebra of Lebesgue measurable ∞  pn where pn is the measure of the subsets of [0, s] and l is Lebesgue measure. Here s = 1 − n=1

point yn . Theorem 2.1. If (X, B, μ, T ) is a m.p.s. where (X, B, μ) is a Lebesgue space, then there is a t.d.s. (Y, S) such that π : (Y, BY , ν, S) → (X, B, μ, T ) is measure-theoretically isomorphic, where ν is an S-invariant Borel probability measure and BY is the Borel σ -algebra of Y . 2.3. Discrete spectrum and almost periodic functions Let (X, B, μ, T ) be a m.p.s., an eigenfunction is some non-zero function f ∈ L2 (X, B, μ) such that Uf = f ◦ T = λf for some λ ∈ C. In this case, λ is called the eigenvalue corresponding to f . It is easy to see every eigenvalue has norm one, that is |λ| = 1. If f ∈ L2 (X, B, μ) is an eigenfunction, then {U n f : n ∈ Z} is precompact in L2 (X, B, μ). Generally, we say that f is almost periodic if {U n f : n ∈ Z} is precompact in L2 (X, B, μ). The set of all almost periodic functions (denoted by Hc ) is spanned by the set of eigenfunctions, and there exists a T -invariant σ -algebra Kμ ⊂ B (the Kronecker factor of (X, B, μ, T )) such that Hc = L2 (X, Kμ , μ). (X, B, μ, T ) is said to have discrete spectrum if L2 (X, B, μ) is spanned by the set of eigenfunctions, that is Hc = L2 (X, B, μ). 3. Complexity function and discrete spectrum Let (X, B, μ, T ) be a m.p.s. and α = {A1 , . . . , Al } be a measurable partition of X. For a point x ∈ X, the α-name α(x) is the bi-infinite sequence {αi (x)}∞ i=−∞ where αi (x) = h whenever i T x ∈ Ah . For x, y ∈ X and n ≥ 1, let 1 Hnα (x, y) = #{0 ≤ i ≤ n − 1 : αi (x) = αi (y)}. n Then Hnα is a sequence of pseudo-metric on X. For x ∈ X, n ≥ 1 and ε > 0, let Bnα (x, ε) = {y ∈ X : Hnα (x, y) < ε} and K(n, α, ε) = min{#(F ) : μ(



x∈F

Bnα (x, ε)) > 1 − ε}.

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It was proved by Ferenczi [2, Proposition 3] that if μ is ergodic, (X, B, μ, T ) has discrete spectrum if and only if for any finite partition α of X and ε > 0 there exists C > 0 such that K(n, α, ε) ≤ C for all n ≥ 1. We will show Ferenczi’s result holds for any m.p.s. without the assumption of ergodicity. Let (X, T ) be a t.d.s. with a metric d, μ ∈ M(X, T ). We say that a compact subset K of X is an equicontinuous set in the mean if for every  > 0, there exists a δ > 0 such that for any n ∈ N and 1 i i any x, y ∈ K with d(x, y) < δ we have n n−1 i=0 d(T x, T y) < . (X, T ) is μ-equicontinuous in the mean if for every τ > 0 there exists a compact subset K of X which is an equicontinuous set in the mean with μ(K) > 1 − τ . Theorem 3.1. Let (X, T ) be a t.d.s. and μ ∈ M(X, T ). Then the following statements are equivalent: (1) (X, T ) is μ-mean equicontinuous. (2) (X, T ) is μ-equicontinuous in the mean. (3) For any finite partition α = {A1 , . . . , Al } of X and ε > 0 there exists C > 0 such that K(n, α, ε) ≤ C for all n ≥ 1. (4) For any finite partition α = {A1 , . . . , Al } of X and ε > 0 there is k ∈ N and measurable k

sets B1 , B2 , · · · , Bk satisfying μ( ∪ Bi ) > 1 − ε, and if x, y ∈ Bi for some 1 ≤ i ≤ k, then lim sup Hnα (x, y) < ε.

i=1

n→∞

(5) For any finite partition α = {A1 , . . . , Al } of X and ε > 0 there is k ∈ N and measurable k

sets B1 , B2 , · · · , Bk satisfying μ( ∪ Bi ) > 1 − ε, and if x, y ∈ Bi for some 1 ≤ i ≤ k, then Hnα (x, y) < ε for any n ∈ N.

i=1

Proof. (1) ⇔ (2) is [7, Theorem 4.2]. (2) ⇒ (3) Let α = {A1 , . . . , Al } be a measurable partition of X. Without loss of generality, we assume that μ(Ai ) > 0 for any 1 ≤ i ≤ l. By the regularity of μ, for any Ai and any m ∈ N there exists a compact subset Pim ⊂ Ai such that μ(Ai \ Pim ) < then μ(Um ) <

1 m

1 m μ(Ai ).

l

Let Um = X \ ∪ Pim , i=1

for m ∈ N. Let sm = min d(Pim , Pjm ). Fix  > 0, there is m ∈ N such that 1≤i =j ≤l

2

μ(Um ) < m1 < 2 . As (X, T ) is μ-equicontinuous in the mean, there is a compact subset K ⊂ X with μ(K) > 2 1 − 2 and δ > 0 such that 1 d(T i x, T i y) <  2 sm n n−1 i=0

for all n ∈ N and x, y ∈ K with d(x, y) < δ. Let U = K ∩ Umc , then μ(U ) > 1 −  2 . For n ∈ N and x ∈ X, let E(x) = {i ≥ 0 : T i x ∈ U } and En = {x ∈ X :

#(E(x) ∩ [0, n − 1]) > 1 − }. n

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Then μ(En ) > 1 − . By the regularity of μ, we can choose compact set Gn ⊂ En ∩ U such that μ(Gn ) > 1 −  −  2 .  i i 2 If x, y ∈ Gn and d(x, y) < δ, then x, y ∈ K and n1 n−1 i=0 d(T x, T y) <  sm . Let An (x, y) = {0 ≤ i ≤ n − 1 : d(T i x, T i y) < sm }, l

then n1 #(An (x, y)) ≥ 1 −  2 . For i ∈ E(x) ∩ E(y) ∩ An (x, y), T i x, T i y ∈ K ∩ ( ∪ Pjm ). By the j =1

l



l

definition of sm , we have (T i x, T i y) ∈ ∪ Pjm × Pjm ⊂ ∪ Aj × Aj . So Hnα (x, y) ≤ 2 +  2 . j =1 j =1  δ Since U is compact, there exist x1 , x2 , · · · , xM ∈ U such that M r=1 B(xr , 2 ) ⊇ U . Let In = δ δ n {r ∈ [1, M] : B(xr , 2 ) ∩ Gn = ∅}. For r ∈ In , we choose yr ∈ B(xr , 2 ) ∩ Gn . Then



 δ B(yrn , δ) ∩ Gn ⊇ B(xr , ) ∩ Gn = Gn . 2

r∈In

So μ(



r∈In

r∈In

Bnα (yrn , 2 +  2 )) > 1 −  −  2 , then K(n, α,  2 + 2) ≤ M for any n ∈ N.

(3) ⇒ (5) Let  > 0. There is C = C(α, ) ∈ N such that for any n ∈ N, we have Fn ⊂ X with #(Fn ) ≤ C and μ



 Bnα (x, /8)

> 1 − /8.

x∈Fn l

Let D = X × X \ ∪ Ai × Ai . By the Birkhoff pointwise ergodic theorem for μ × μ a.e. (x, y) ∈ i=1

X2 1 1D (T i x, T i y) = H ∗ (x, y). n→∞ n n−1

lim Hnα (x, y) = lim

n→∞

i=0

 So for a given 0 < r < min{1, 4C }, by Egorov’s theorem there are R ⊂ X 2 with μ × μ(R) > 1 − r 2 and N ∈ N such that if (x, y) ∈ R then

|Hnα (x, y) − HNα (x, y)| < r, for n ≥ N. By Fubini’s theorem there is A ⊂ X such that μ(A) > 1 − r and for any x ∈ A, μ(Rx ) > 1 − r, where Rx = {y ∈ X : (x, y) ∈ R}. α (x , /8) = ∅}. Enumerate FN = {x1 , x2 , . . . , xm }. Then m ≤ C. Let I = {1 ≤ i ≤ m : A ∩ BN i α

Denote #(I ) = m . Then 1 ≤ m ≤ m. For each i ∈ I , pick yi ∈ A ∩ BN (xi , /8). Then we have α (x , /8) ⊂ B α (y , /4) for all i ∈ I . As BN i N i

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  α μ A∩ Ryi ∩ BN (x, /8) ≥ 1 − r − m r − /8 > 1 − , i∈I

x∈FN

choose a compact subset K ⊂A∩



Ryi ∩

i∈I



α BN (x, /8) = A ∩

x∈FN



Ryi ∩

i∈I



α BN (x, /8)

x∈I

α (y , /4). Then (y , x) ∈ R. with μ(K) > 1 −. If x ∈ K, there exists i ∈ I such that x ∈ Ryi ∩BN i i By the construction of R, for any n ≥ N ,

Hnα (yi , x) ≤ HNα (yi , x) + r < /4 + r < /2. Let

N −1 i=0

Then K =



α (y , /4) ∩ K. T −i α = {C1 , C2 , · · · , Cp }, then p ≤ l N . Let Eji = Cj ∩ Ryi ∩ BN i p

∪ Eji . For any (x, y) ∈ Eji , we have Hnα (x, y) = 0 for any 1 ≤ n ≤ N − 1, thus

i∈I j =1

Hnα (x, y) <  for any n ∈ N. (5) ⇒ (2) Assume that diam(X) ≤ 1. For ε > 0 and m ∈ N there is finite partition α m = {A1m , A2m , · · · , Atmm } of X with diam(Aim ) < 2m . There is nm ∈ N and measurable sets nm

1 , B 2 , · · · , B nm satisfying μ( ∪ B i ) > 1 − Bm m m m i=1

m

 2m ,

i for 1 ≤ i ≤ n then such that if x, y ∈ Bm m

 i 2m for any n ≥ 1. By the regularity of μ, we can assume that Bm is nm ∞ i , then μ(K ) > 1 −  . Let K = ∩ K , then μ(K) > 1 − ε. Km = ∪ Bm m m 2m i=1 m=1

Hnα (x, y) < Let

a compact set.

Now we show K is equicontinuous in the mean. Suppose to the contrary that K is not equicontinuous in the mean. Then by definition there exists η > 0 such that for any k ≥ 1 there are xk , yk ∈ K and sk ∈ N such that d(xk , yk ) < 1k and d¯sk (xk , yk ) ≥ η. As K is compact, without loss of generality assume that xk → x ∈ K, we also have yk → x. For any k ∈ N, by the triangle inequality, either d¯sk (xk , x) ≥ η2 or d¯sk (yk , x) ≥ η2 . Without loss of generality, we always have d¯sk (xk , x) ≥ η2 . k = {0 ≤ i ≤ s −1 : α m (x) = α m (x )}. Then H α m (x , x) = 1 #(E k ). Since diam(X) ≤ Let Em k k k sk m i i sk k then d(T i x, T i x ) ≤ 1, if i ∈ [0, s − 1] \ E k then 1 and diam(Aim ) < 2m , if i ∈ Em k k m d(T i x, T i xk ) ≤ 2m . So for any m ∈ N and any k ∈ N 1  η k k (#(Em ) × 1 + (sk − #(Em )) × m ) ≥ d¯sk (xk , x) ≥ . sk 2 2 Fix m ∈ N with



2m

m

k ) > η s . Then H α (x , x) > η . Since {x }∞ ⊂ K, < η4 , we have #(Em

k k k=1 sk 4 k 4

i ∞ {xk }∞ k=1 ⊂ Km . Without loss of generality, we assume {xk }k=1 ⊂ Bm for some 1 ≤ i ≤ nm . m

i is compact, x ∈ B i , then H α (x , x) < Since Bm

k sk m (5) ⇒ (3) and (5) ⇒ (4) are obvious.



2m

< η4 , a contradiction. k

(4) ⇒ (5) Fix  > 0, then there exist measurable sets B1 , B2 , · · · , Bk satisfying μ( ∪ Bi ) > i=1

1−

ε 8

and lim sup n→∞

Hnα (x, y)

k

< ε/8 for (x, y) ∈ ∪ Bi × Bi . Let i=1

H ∗ (x, y)

= lim sup Hnα (x, y), n→∞

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l

then H ∗ is a pseudo-metric on X. Let D = X × X \ ∪ Ai × Ai . By the Birkhoff pointwise i=1

ergodic theorem for μ × μ a.e. (x, y) ∈ X 2

1 1D (T i x, T i y) = H ∗ (x, y). n→∞ n n−1

lim Hnα (x, y) = lim

n→∞

i=0

 So for a given 0 < r < min{1, 4k }, by Egorov’s theorem there are R ⊂ X 2 with μ × μ(R) > 2 1 − r and N ∈ N such that if (x, y) ∈ R then

|Hnα (x, y) − H ∗ (x, y)| < r, for n ≥ N. By Fubini’s theorem there is A ⊂ X such that μ(A) > 1 − r and for any x ∈ A, μ(Rx ) > 1 − r, where Rx = {y ∈ X : (x, y) ∈ R}. Let I = {1 ≤ i ≤ k : A ∩ Bi = ∅}. Denote #(I ) = k . Then 1 ≤ k ≤ k. Let B ∗ (x, δ) = {y ∈ X : H ∗ (x, y) < δ}. For each i ∈ I , pick yi ∈ A ∩ Bi . Then we have Bi ⊂ B ∗ (yi , /8) for all i ∈ I . As  μ A∩



Ryi ∩

i∈I

k



Bi ≥ 1 − r − k r − /8 > 1 − ,

i=1

choose a compact subset K ⊂A∩

i∈I

Ryi ∩

k i=1

Bi = A ∩

i∈I

Ryi ∩



Bi

i∈I

with μ(K) > 1 − . If x ∈ K, there exists i ∈ I such that x ∈ Ryi ∩ B ∗ (yi , /8). Then (yi , x) ∈ R. By the construct of R, for any n ≥ N , Hnα (yi , x) ≤ H ∗ (yi , x) + r < /8 + r < /2.

N −1

T −i α = {C1 , C2 , · · · , Cp }, then p ≤ l N . Let Eji = Cj ∩ Ryi ∩ B ∗ (yi , /8) ∩ K.  p ∪ Eji . For any (x, y) ∈ Eji , we have Hnα (x, y) = 0 for 1 ≤ n ≤ N − 1, thus Then K = Let

i=0

i∈I j =1

Hnα (x, y) <  for any n ∈ N.

2

Corollary 3.2. Let (X, B, μ, T ) be a m.p.s. where (X, B, μ) is a Lebesgue space. Then the following statements are equivalent: (1) (X, B, μ, T ) has discrete spectrum. (2) For any finite partition α = {A1 , . . . , Al } of X and ε > 0 there exists C ∈ N such that K(n, α, ε) ≤ C for any n ≥ 1.

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Proof. By Theorem 2.1, there is a topological dynamical system (Y, S) such that π : (Y, BY , ν, S) → (X, B, μ, T ) is measure-theoretically isomorphic, where ν is an S-invariant Borel probability measure and BY is the Borel σ -algebra of Y . (1) ⇒ (2) Then (Y, BY , ν, S) has discrete spectrum. By [7, Theorem 4.2, 4.3, 4.6], (Y, BY , ν, S) is μ-mean equicontinuous. By Theorem 3.1 for any finite partition π −1 α = {π −1 A1 , . . . , π −1 Al } on (Y, BY , ν, S) and ε > 0 there exists C ∈ N such that K(n, π −1 α, ε) ≤ C for all n ≥ 1. Since π is measure-theoretically isomorphic, K(n, α, ε) ≤ C for any n ≥ 1. (2) ⇒ (1) follows the same idea as (1) ⇒ (2). 2 4. Measure-theoretic mean equicontinuous functions and complexity functions 4.1. Measure-theoretic mean equicontinuous functions In this subsection, we will give two definitions about measure-theoretic mean equicontinuous functions, and show that they are equivalent. Definition 4.1. Let (X, T ) be a t.d.s. with a metric d, μ ∈ M(X, T ) and f ∈ L1 (X, μ). We say that a compact subset K of X is an f -mean equicontinuous set if for every  > 0, there exists a δ > 0 such that 1 |f (T i x) − f (T i y)| <  n n−1

lim sup n→∞

i=0

for all x, y ∈ K with d(x, y) < δ. We say that f is (μ, d)-mean equicontinuous if for every τ > 0 there exists an f -mean equicontinuous set K of X with μ(K) > 1 − τ . Definition 4.2. Let (X, B, μ, T ) be a measure preserving system. We say a function f ∈ L1 (X, μ) is μ-mean equicontinuous if for any  > 0 there is k ∈ N and measurable sets k

A1 , A2 , · · · , Ak satisfying μ( ∪ Ai ) > 1 − , and if x, y ∈ Ai for some 1 ≤ i ≤ k then i=1

1 lim sup |f (T i x) − f (T i y)| < . n→∞ n n−1 i=0

Proposition 4.3. Let (X, T ) be a t.d.s. with a metric d, μ ∈ M(X, T ) and f ∈ L1 (X, μ). Then the following statements are equivalent: (1) f is (μ, d)-mean equicontinuous. (2) f is μ-mean equicontinuous. Proof. (1) ⇒ (2) For any  > 0, there is a compact subset M ⊂ X with μ(M) > 1 − , and there is δ > 0 whenever d(x, y) < 2δ with x, y ∈ M 1 |f (T i x) − f (T i y)| < . n n−1

lim sup n→∞

i=0

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By the compactness of M, there is k ∈ N and x1 , . . . , xk ∈ M such that M = ∪ B(xi , δ) ∩ M. i=1

k

Let Ai = B(xi , δ) ∩ M for 1 ≤ i ≤ k, then μ( ∪ Ai ) > 1 − , thus f is μ-mean equicontinuous. i=1

(2) ⇒ (1) For τ > 0 and any n ∈ N, there exist kn pairwise disjoint measurable subsets kn

n with μ( ∪ Ani ) > 1 − {Ani }ki=1

i=1

τ 2n

such that if x, y ∈ Ani ,

lim sup m→∞

m−1 1  1 |f (T i x) − f (T i y)| < . m n i=0

By the regularity of μ, we can require Ani to be compact. Let δn =

min d(Ani , Anj ). Let

1≤i =j ≤kn

∞

kn

Mn = ∪ Ani , M = ∩ Mn , then μ(M) > 1 − τ and M is a compact set. For any  > 0 there is i=1

N ∈ N such that x, y ∈ AN i . Then

n=1

1 N

< . Then whenever d(x, y) < δN with x, y ∈ M, there exists AN i such that

lim sup m→∞

m−1 1  1 < . |f (T i x) − f (T i y)| < m N i=0

Then M is an f -mean equicontinuous set, f is (μ, d)-mean equicontinuous.

2

Remark 4.4. This proposition shows that for a t.d.s. (X, T ), μ ∈ M(X, T ), the definition of (μ, d)-mean equicontinuous functions does not rely on the metric d. Thus in the following, we will simply use μ-mean equicontinuous functions instead of (μ, d)-mean equicontinuous functions. Definition 4.5. Let (X, T ) be a t.d.s. with a metric d, μ ∈ M(X, T ) and f ∈ L1 (X, μ). We say that a compact subset K of X is f -equicontinuous in the mean if for any  > 0 there is δ > 0 whenever d(x, y) < δ with x, y ∈ K, we have 1 |f (T i x) − f (T i y)| < , n n−1 i=0

for any n ≥ 1. We say f is μ-equicontinuous in the mean if for any τ > 0 there is a compact subset K which is f -equicontinuous in the mean with μ(K) > 1 − τ . Theorem 4.6. Let (X, T ) be a t.d.s. with a metric d, μ ∈ M(X, T ) and f ∈ L1 (X, μ). Then the following statements are equivalent: (1) f is μ-equicontinuous in the mean. (2) f is μ-mean equicontinuous.

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Proof. (1) ⇒ (2) is obvious. (2) ⇒ (1) Fix ε > 0, there is a compact K ⊂ X such that μ(K) > 1 −  and K is an f -mean equicontinuous set. For any l ∈ N there exists a δl > 0 such that 1 1 |f (T i x) − f (T i y)| < n l n−1

lim sup n→∞

i=0

for allx, y ∈ K with d(x, y) < δl . As K is compact, there exists a finite subset Fl of K such that K ⊂ x∈Fl B(x, δl ). Enumerate Fl as {xl1 , xl2 , . . . , xlml }. For j = 1, . . . , ml and N ∈ N, let j AN l (xl ) =

  n−1 1 1 j i j i y ∈ B(xl , δl ) ∩ K : |f (T xl ) − f (T y)| < , n = N, N + 1, . . . . n l i=0

∞ It is easy to see that for each j = 1, . . . , ml , {AN l (xl )}N =1 is an increasing sequence and j

∞

j

j

B(xl , δl ) ∩ K = ∪ AN l (xl ). Choose Nl ∈ N and a compact subset Kl of K such that μ(Kl ) > N =1

μ(K) −

 2l

j

j

l with Kl ∩ B(xl , δl ) ⊂ AN l (xl ) for j = 1, . . . , ml .

∞

j

Let K0 = ∩ Kj , then μ(K0 ) > 1 − 2. Let δl,1 be the Lebesgue number of {B(xl , δl ) : j = j =1

1, · · · , ml }. Then for n = Nl , Nl + 1, · · · and every x, y ∈ K0 with d(x, y) < δl,1 , 1 2 |f (T i x) − f (T i y)| < . n l n−1 i=0

By Lusin’s Theorem, there exists a compact set Al ⊂ K0 with μ(Al ) > μ(K0 ) − is continuous on Al for 0 ≤ i ≤ Nl − 1. Then there exists δl,2 > 0 such that

 2l

and f ◦ T i

1 1 |f (T i x) − f (T i y)| < n l n−1 i=0

for n = 1, 2, · · · , Nl − 1 and every x, y ∈ Al with d(x, y) < δl,2 . Let δl,3 = min{δl,1 , δl,2 }. Then by the construction, we know that for every x, y ∈ Al with d(x, y) < δl,3 , we have ∞ 1 n−1 2 i i i=0 |f (T x) − f (T y)| < l for n ∈ N. Let A = ∩ Al , then μ(A) > 1 − 3 and A is n l=1

f -equicontinuous in the mean, f is μ-equicontinuous in the mean. 2 4.2. Bounded complexity with respect to functions In this subsection, we study measure-theoretic mean equicontinuous functions and measuretheoretic complexity with respect to some functions. Let (X, T ) be a t.d.s. and μ ∈ M(X, T ), f ∈ L1 (X, μ). Define n−1   1 |f (T i x) − f (T i y)| and fˆn (x, y) = max f¯k (x, y) : 1 ≤ k ≤ n . f¯n (x, y) = n i=0

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Then f¯n and fˆn are pseudo-metrics on X. For x ∈ X, n ≥ 1 and ε > 0, let Bf¯n (x, ε) = {y ∈ X : f¯n (x, y) < ε}. Bfˆn (x, ε) = {y ∈ X : fˆn (x, y) < ε}, and K(n, ε, f¯) = min{#(F ) : μ(



Bf¯n (x, ε)) > 1 − ε}.

x∈F

K(n, ε, fˆ) = min{#(F ) : μ(



Bfˆn (x, ε)) > 1 − ε}.

x∈F

We say that μ has bounded complexity with respect to {f¯n } (resp. {fˆn }) if for every  > 0 there exists a positive integer C = C() such that K(n, ε, f¯) ≤ C (resp. K(n, ε, fˆ) ≤ C) for all n ≥ 1. Motivated by [7, Theorem 4.2, 4.3], we have the following theorem. Theorem 4.7. Let (X, T ) be a t.d.s. and μ ∈ M(X, T ), f ∈ L1 (X, μ). Then the following statements are equivalent: (1) (2) (3) (4)

f is μ-mean equicontinuous. f is μ-equicontinuous in the mean. μ has bounded complexity with respect to {f¯n }. μ has bounded complexity with respect to {fˆn }. k

(5) For any  > 0 there is k ∈ N and measurable subsets A1 , · · · , Ak with μ( ∪ Ai ) > 1 −  i=1

and fˆn (x, y) <  for any x, y ∈ Ai and any n ∈ N.

Proof. (1) ⇔ (2) is Theorem 4.6. (2) ⇒ (5) Fix ε > 0, there exists a compact subset K of X with μ(K) > 1 − ε such that K is f -equicontinuous in the mean. There exists δ > 0 such that for x, y ∈ K with d(x, y) < δ and any n ∈ N 1 |f (T i x) − f (T i y)| < , f¯n (x, y) = n n−1 i=0

ˆn (x, y) <  for any n ∈ N. As K is compact, there exists a finite subset F of K such that thus f K ⊂ x∈F B(x, 2δ ). Enumerate F as {x1 , x2 , . . . , xm }. Then fˆn (x, y) <  for any n ≥ 1 with x, y ∈ B(xi , 2δ ) ∩ K, 1 ≤ i ≤ m. (5) ⇒ (2) For  > 0 and any m ∈ N there is km ∈ N and measurable subsets {Aim , i = km

1, 2, · · · , km } with μ( ∪ Aim ) > 1 − i=1

 2m

and fˆn (x, y) <

 2m

for any x, y ∈ Aim and any n ∈ N. km

∞

i=1

m=1

By the regularity of μ, we can assume Aim to be compact. Let Km = ∪ Aim and K = ∩ Km , then μ(K) > 1 − .

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We show K is f -equicontinuous in the mean. Assume that K is not f -equicontinuous in the mean. Then there exists τ > 0 for any k ≥ 1, there are xk , yk ∈ K and mk ∈ N such that d(xk , yk ) < 1k and fˆmk (xk , yk ) ≥ τ . As K is compact, without loss of generality assume that xk → x ∈ K, we also have yk → x. For any k ∈ N, by the triangle inequality, either fˆmk (xk , x) ≥ τ τ τ ˆ ˆ 2 or fmk (yk , x) ≥ 2 . Without loss of generality, we always assume fmk (xk , x) ≥ 2 .  τ ∞ ∞

Choose m ∈ N such that m < 2 . Since {xk }k=1 ⊂ K, {xk }k=1 ⊂ Km . Without loss of gen2

i i erality, we assume that {xk }∞ k=1 ⊂ Am for some 1 ≤ i ≤ km . As Am is closed, x is also in Aim . Note that for any u, v ∈ Aim and any n ≥ 1, fˆn (u, v) < m < τ2 . Particularly, we have 2 fˆmk (xk , x) <  < τ , a contradiction. 2m

2

(5) ⇒ (4) and (4) ⇒ (3) are obvious. (3) ⇒ (5) Assume that μ has bounded complexity with respect to {f¯n }. Let  > 0. There is C = C() such that for any n ∈ N, there is Fn ⊂ X with #(Fn ) ≤ C such that μ



 Bf¯n (x, /8) > 1 − /8.

x∈Fn

By the Birkhoff pointwise ergodic theorem for μ × μ a.e. (x, y) ∈ X 2 1 |f (T i x) − f (T i y)| → f ∗ (x, y). f¯n (x, y) = n n−1 i=0

 So for a given 0 < r < min{1, 4C }, by Egorov’s theorem there are R ⊂ X 2 with μ × μ(R) > 2 1 − r and N ∈ N such that if (x, y) ∈ R then

|f¯n (x, y) − f¯N (x, y)| < r, for n ≥ N. By Fubini’s theorem there is A ⊂ X such that μ(A) > 1 − r and for any x ∈ A, μ(Rx ) > 1 − r, where Rx = {y ∈ X : (x, y) ∈ R}. Enumerate FN = {x1 , x2 , . . . , xm }. Then m ≤ C. Let I = {1 ≤ i ≤ m : A ∩ Bf¯N (xi , /8) = ∅}. Denote #(I ) = m . Then 1 ≤ m ≤ m. For each i ∈ I , pick yi ∈ A ∩ Bf¯N (xi , /8). Then we have Bf¯N (xi , /8) ⊂ Bf¯N (yi , /4) for all i ∈ I . As   μ A∩ Ryi ∩ Bf¯N (x, /8) ≥ 1 − r − m r − /8 > 1 − , i∈I

x∈FN

for any i ∈ I , by Lusin’s Theorem we can choose compact subset Ki ⊂ A ∩



Ryi ∩ Bf¯N (xi , /8)

i∈I

with μ( ∪ Ki ) > 1 −  and f ◦ T j is continuous on ∪ Ki for 0 ≤ j ≤ N . Let K = ∪ Ki . i∈I

i∈I

i∈I

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If x ∈ K, there exists i ∈ I such that x ∈ Ki ⊂ Ryi ∩ Bf¯N (yi , /4). Then (yi , x) ∈ R. By the construct of R, for any n ≥ N , f¯n (yi , x) ≤ f¯N (yi , x) + r < /4 + r < /2. Since f ◦ T j is continuous on K for 0 ≤ j ≤ N , there is δ > 0 such that for any x1 , x2 ∈ K with d(x1 , x2 ) < δ, we have |f (T j x1 ) − f (T j x2 )| <  for any 0 ≤ j ≤ N . Thus f¯n (x1 , x2 ) <  for1 ≤ n ≤ N . Since Ki is compact, there exists a finite subset Hi of Ki such that Ki ⊂ B(x, 2δ ). Then for any x1 , x2 ∈ B(x, 2δ ) ∩ Ki where x ∈ Hi , we have x∈Hi

¯ for any f¯n (x1 , x2 ) ≤ f¯n (x1 , yi ) + f¯n (x2 , yi ) <    n ≥ Nδ , and fn (x1 , x2 ) <  for 1 ≤ n ≤ Nδ . Thus ˆ fn (x1 , x2 ) <  for any n ≥ 1. Thus K ⊂ ( B(x, 2 ) ∩ Ki ), and for any x1 , x2 ∈ B(x, 2 ) ∩ Ki i∈I x∈Hi

where x ∈ Hi , fˆn (x1 , x2 ) <  for any n ≥ 1.

2

5. Measure-theoretic mean equicontinuous function and almost periodic function García-Ramos and Marcus [6, Corollary 3.8] proved measure-theoretic mean equicontinuous function and almost periodic function are equivalent for ergodic measure. Inspired by [9, Proposition 4.1] and [13, Theorem 6, Theorem 7], we show the equivalence for invariant measure. To prove the equivalence, we need the following lemma. Lemma 5.1. Let (Xi , Bi , μi , Ti ) be measure preserving systems, fi ∈ L2 (Xi , μi ), i = 1, 2. Let F (x, y) = f1 (x) − f2 (y), then the following statements are equivalent: (1) fi is an almost periodic function in L2 (Xi , Bi , μi , Ti ), i = 1, 2. (2) F is an almost periodic function in L2 (X1 × X2 , B1 × B2 , μ1 × μ2 , T1 × T2 ). Proof. Let Ki ⊂ Bi be the Kronecker factor of (Xi , Bi , μi , Ti ). Let Hi = L2 (Xi , Bi , μi ), Hi = L2 (Xi , Ki , μi ) and Hi = Hi ⊕ Hi

be the orthogonal compositions. Since L2 (X1 × X2 , B1 × B2 , μ1 × μ2 ) = H1 ⊗ H2 = (H1 ⊗ H2 ) ⊕ (H1 ⊗ H2

) ⊕ (H1

⊗ H2 ) ⊕ (H1

⊗ H2

). By [4, Lemma 4.17] (H1 ⊗ H2 ) consists of all the almost periodic functions of L2 (X1 × X2 , B1 × B2 , μ1 × μ2 , T1 × T2 ), i.e. K1 × K2 ⊂ B1 × B2 is the Kronecker factor of (X1 × X2 , B1 × B2 , μ1 × μ2 , T1 × T2 ). (1) ⇒ (2) If fi is an almost periodic function in L2 (Xi , Bi , μi , Ti ), i = 1, 2, then fi ∈ Hi . It is easy to see that f1 (x), f2 (y) ∈ H1 ⊗ H2 , so F (x, y) = f1 (x) − f2 (y) ∈ H1 ⊗ H2 . (2) ⇒ (1) Let fi = fi + fi

be the orthogonal compositions such that fi ∈ Hi , fi

∈ Hi

. Then f1 (x) ∈ H1 ⊗ H2 , f1

(x) ∈ H1

⊗ H2 , f2 (y) ∈ H1 ⊗ H2 , f2

(y) ∈ H1 ⊗ H2

. Since F is an almost periodic function in L2 (X1 × X2 , B1 × B2 , μ1 × μ2 , T1 × T2 ),

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F (x, y) = f1 (x) − f2 (y) = (f1 (x) − f2 (y)) + f1

(x) − f2

(y) ∈ (H1 ⊗ H2 ). So f1

(x) = 0 and f2

(y) = 0, fi ∈ Hi , i = 1, 2 and fi is an almost periodic function in L2 (Xi , Bi , μi , Ti ). 2 Theorem 5.2. Let (X, T ) be a t.d.s., μ ∈ M(X, T ) and f ∈ L2 (X, μ). Then the following statements are equivalent: (1) f is an almost periodic function in L2 (X, μ). (2) f is μ-mean equicontinuous. Proof. (2) ⇒ (1) Suppose that f is not almost periodic in L2 (X, μ), for R > 0 define  f (x) = R

f (x), |f (x)| ≤ R, f (x) R |f (x)| , |f (x)| > R

then f R − f L2 (X) → 0 when R → ∞. By [14, Theorem 7.1] there exists R > 0 such that f R is not almost periodic in L2 (X, μ). Since |f (x) − f (y)| ≥ |f R (x) − f R (y)|, then lim sup fˆn (x, y) ≥ lim sup fˆnR (x, y). n→∞

n→∞

R

Since f is μ-mean equicontinuous, f R is also μ-mean equicontinuous. Let h(x) = f 4R(x) , then |h(x)| ≤ 14 , h is μ-mean equicontinuous and h is not almost periodic in L2 (X, μ). Let H (x, y) = h(x) − h(y), then H ∈ L2 (X × X, μ × μ). Since h is not almost periodic in L2 (X, μ), by Lemma 5.1 H is not almost periodic in L2 (X × X, μ × μ). Let Hn (x, y) = H (T n x, T n y), then there exists c1 > 0 and an increasing sequence of positive integers p1 , p2 , · · · such that Hpi − Hpj L2 (X2 ) ≥ c1 for any i = j ∈ N. Since |H (x, y)| = |h(x) − h(y)| ≤ 12 , then there exists c = (c1 )2 > 0 such that Hpi − Hpj L1 (X2 ) ≥ c. c Choose  = 18 , by Theorem 4.7 there is k ∈ N and measurable subsets A1 , · · · , Ak with k

k

i=1

i=1

μ( ∪ Ai ) > 1 −  and hˆ n (x, y) <  for any x, y ∈ Ai and any n ∈ N. Let A0 = X \ ∪ Ai . Choose points ai in Ai arbitrarily for i = 1, · · · , k. For n ∈ N, let  gn (x, y) =

0, x ∈ A0 or y ∈ A0 , Hn (ai , aj ), x ∈ Ai , y ∈ Aj (1 ≤ i, j ≤ k).

Let A00 = (A0 × X) ∪ (X × A0 ), since μ(A0 ) < , μ × μ(A00 ) < 2. On Ai × Aj we have |Hn (x, y) − gn (x, y)| = |Hn (x, y) − Hn (ai , aj )| = |h(T n x) − h(T n y) − h(T n ai ) + h(T n aj )| ≤ |h(T n x) − h(T n ai )| + |h(T n y) − h(T n aj )| = |Hn (x, ai )| + |Hn (y, aj )|.

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Then 

n−1 

|Hm (x, y) − gm (x, y)|dμ × μ

X×X m=0

=

k 



n−1 

|Hm (x, y) − gm (x, y)|dμ × μ +

i,j =1A ×A m=0 i j

≤

k 



n−1 

k 

|Hm (x, y) − gm (x, y)|dμ × μ

A00 m=0

(|Hm (x, ai )| + |Hm (y, aj )|)dμ × μ +

i,j =1A ×A m=0 i j

<

  n−1

  n−1

A00

 n(hn (x, ai ) + hn (y, aj ))dμ × μ +

i,j =1A ×A i j

|Hm (x, y)|dμ × μ

m=0

n × 2 < 3n 2

Since 3n = nc 6 , there exists finite set Fn ⊂ [0, n − 1] with #(Fn ) ≥ gm L1 (X2 ) < 3c holds for m ∈ Fn . Let

L = {g ∈ L1 (X 2 ) : g =

k  i,j =1

n 2

such that Hm −

1 cij 1Ai ×Aj , |cij | ≤ }, 2

then {gn : n ∈ N} ⊂ L and L has dimension k 2 . Then there exist C = C(c) ∈ N and Fn = n a b

n) a {tn1 , tn2 , . . . , tnmn } ⊂ Fn with mn ≥ #(F C ≥ 2C such that for any tn , tn ∈ Fn we have gtn − c gtnb L1 (X2 ) ≤ 3 , then Htna − Htnb L1 (X2 ) < c. Let M = 6C and choose N ≥ pM , then tNi + pk ≤ 2N for i = 1, 2, · · · , mN , k = 1, 2, · · · , M. N But mN × M ≥ 2C × 6C = 3N > 2N , there exists a = b ∈ [1, mN ], u = v ∈ [1, M] such a b that tN + pu = tN + pv . Then HtNa − Ht b L1 (X2 ) = Hpu − Hpv L1 (X2 ) . However HtNa − N Ht b L1 (X2 ) < c and Hpu − Hpv L1 (X2 ) ≥ c, a contradiction. N ∞  (1) ⇒ (2) Fix  > 0. Choose t > 0 such that t < . Since the collection of almost periodic t=1

function is spanned by the set of eigenfunctions of L2 (X, μ), we can find eigenfunctions {hk }∞ k=1 of L2 (X, μ) and at,k ∈ C for k = 1, 2, · · · , Kt such that

f −

Kt 

at,k hk L2 (X,μ) < t2 .

k=1

 t 2 Then f − K k=1 at,k hk L1 (X,μ) < t . For k ∈ N, there exist λk ∈ C with |λk | = 1, and Xk ∈ BX with μ(Xk ) = 1, T Xk = Xk such that hk (T x) = λk hk (x) for all x ∈ Xk . Let

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gt (x) = |f (x) −

Kt 

at,k hk (x)|,

k=1

 then X gt dμ < t2 . By the Birkhoff pointwise ergodic theorem, there exists Xt ∈ BX with μ(Xt ) = 1 such that 1 gt (T i x) = gt∗ (x) n→∞ n n−1

lim

i=0



for x ∈ Xt and

∗ 2 X gt dμ < t .

∞

∞

k=1

t=1

Set X0 = ( ∩ Xk ) ∩ ( ∩ Xt ), then μ(X0 ) = 1 and for x, y ∈ X0 ,

1 |f (T i x) − f (T i y)| n n−1

lim sup n→∞

1 n→∞ n

≤ lim

i=0

n−1 

|f (T i x) −

i=0

Kt 

t  1 |f (T i y) − at,k hk (T i y)| n→∞ n

n−1

K

i=0

k=1

at,k hk (T i x)| + lim

k=1

t 1  | at,k (hk (T i x) − hk (T i y))| n→∞ n

n−1

K

+ lim

i=0 k=1

≤ gt∗ (x) + gt∗ (y) + |

Kt 

at,k (hk (x) − hk (y))|

k=1

Put Yt = {x ∈ X : gt∗ (x) ≥ t }. Since gt∗ L1 (X,μ) < t2 , μ(Yt ) < t . By Lusin’s Theorem there exists a compact subset At of X0 \ Yt such that μ(At ) > 1 − t and hk |At is continuous for k = 1, 2, · · · , Kt . By the continuity of hk |At , k = 1, 2, · · · , Kt , there exists δt > 0 such that when x, y ∈ At with d(x, y) < δt , one has Kt 

|at,k | · |hk (x) − hk (y)| < t .

k=1

Then for x, y ∈ At with d(x, y) < δt , lim sup n→∞

t  1 |f (T i x) − f (T i y)| ≤ gt∗ (x) + gt∗ (y) + | at,k (hk (x) − hk (y))| < 3t . n

n−1

K

i=0

k=1

∞

Let A = ∩ At , then μ(A) > 1 − , and A is an f -mean equicontinuous set. Since  is arbit=1

trary, f is μ-mean equicontinuous.

2

Corollary 5.3. Let (X, B, μ, T ) be a m.p.s. where (X, B, μ) is a Lebesgue space, f ∈ L2 (X, μ). Then the following statements are equivalent:

T. Yu / J. Differential Equations 267 (2019) 6152–6170

6169

(1) f is an almost periodic function in L2 (X, μ). (2) f is μ-mean equicontinuous. (3) μ has bounded complexity with respect to {f¯n }. Proof. Since (X, B, μ) is a Lebesgue space, by Theorem 2.1 there is a t.d.s. (Y, S) with ν an S-invariant Borel probability measure such that π : (X, B, μ, T ) → (Y, BY , ν, S) is measuretheoretically isomorphic. (2) ⇒ (1) By Definition 4.2 f π −1 ∈ L2 (Y, ν) is ν-mean equicontinuous. By Theorem 5.2 f π −1 is an almost periodic function in L2 (Y, ν). So f is an almost periodic function in L2 (X, μ). (1) ⇒ (2) follows the same idea as (2) ⇒ (1). (2) ⇒ (3) Let g = f π −1 , by Definition 4.2 g ∈ L2 (Y, ν) is ν-mean equicontinuous. By Theorem 4.7 ν has bounded complexity with respect to {g¯ n }, so μ has bounded complexity with respect to {f¯n }. (3) ⇒ (2) follows the same idea as (2) ⇒ (3). 2 We will show the relationship between measure-theoretic mean equicontinuity and measuretheoretic mean equicontinuous functions which generalized García-Ramos and Marcus’s result [6, Theorem 3.9] from ergodic measure cases to invariant measure cases. Theorem 5.4. Let (X, T ) be a t.d.s. and μ ∈ M(X, T ). Then the following statements are equivalent: (1) (2) (3) (4)

(X, T ) is μ-mean equicontinuous. For any f ∈ C(X), f is μ-mean equicontinuous. For any f ∈ L2 (X, μ), f is μ-mean equicontinuous. For any B ∈ BX , 1B is μ-mean equicontinuous.

Proof. (1) ⇔ (3) By [7, Theorem 4.2, 4.3, 4.6], (X, T ) is μ-mean equicontinuous if and only if (X, T ) has discrete spectrum. By Theorem 5.2 f is μ-mean equicontinuous if and only if f is an almost periodic function. By definition (X, T ) has discrete spectrum if and only if f is an almost periodic function for any f ∈ L2 (X, μ). (3) ⇒ (4) and (3) ⇒ (2) are obvious. (2) ⇒ (3) By Theorem 5.2, f is μ-mean equicontinuous if and only if f is an almost periodic function in L2 (X, μ). Since C(X) is dense in L2 (X, μ) and almost periodic functions are closed in L2 (X, μ), for any f ∈ L2 (X, μ), f is μ-mean equicontinuous. (4) ⇒ (3) By Theorem 5.2, f is μ-mean equicontinuous if and only if f is an almost periodic function in L2 (X, μ). Since span{1B : B ∈ BX } is dense in L2 (X, μ) and the set of all almost periodic functions Hc is a linear space and closed in L2 (X, μ), for any f ∈ L2 (X, μ), f is μ-mean equicontinuous. 2 References [1] F. Blanchard, B. Host, A. Maass, Topological complexity, Ergod. Theory Dyn. Syst. 20 (3) (2000) 641–662. [2] S. Ferenczi, Measure-theoretic complexity of ergodic systems, Isr. J. Math. 100 (1) (1997) 189–207. [3] S. Fomin, On dynamical systems with a purely point spectrum, Dokl. Akad. Nauk SSSR 77 (1951) 29–32 (in Russian). [4] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, NJ, 1981.

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