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A characterization of {\mu}-equicontinuity for topological dynamical systems

Two different notions of {\mu}-equicontinuity that apply to topological dynamical systems and probability measures were studied by Gilman (1987) and Huang-Lu-Ye (2011). One was used to classify measure preserving topological dynamical systems and the other for cellular automata. We show that if the probability space satisfies Lebesgue's density theorem and Vitali's covering theorem (for example a Cantor set or a subset of R^{d}) then both properties are equivalent. To do this we characterize when a function is Lusin measurable.


Introduction
A topological dynamical system (TDS), (X, T ), is a continuous transformation T on a compact metric space X. A class of rigid systems is the family of equicontinuous TDS. A TDS is equicontinuous if the family {T i } is equicontinuous.
Equivalently we can define equicontinuity using equicontinuity points: x ∈ X is an equicontinuity point if for every y close to x we have that T i (x), T i (y) stay close for all i. A TDS is equicontinuous if every point is equicontinuous. Sensitivity (or sensitivity to initial conditions) is considered a weak form of chaos. Akin-Auslander-Berg [1] showed that a minimal TDS is either sensitive or equicontinuous. .
Equicontinuity is a very strong property and different attempts have been made to weaken this property. In particular we pay attention to the ones made with the use of Borel probability measures.
While studying cellular automata (a subclass of TDS), Gilman [11] [10] introduced the concept of µ−equicontinuity points: x ∈ X is a µ−equicontinuity point if for most y close to x we have that T i (x), T i (y) stay close for all i ∈ Z + . He also defined µ−sensitivity and he showed that if T is a cellular automaton and µ a shift−ergodic measure (not necessarily invariant with respect to T ) then T is either µ−sensitive or the set of µ−equicontinuity points has measure one. With exactly the same approach one can define µ−equicontinuity points for TDS with respect to measures that are not necessarily ergodic or invariant under any transformation.
In [4], Cadre-Jacob introduced µ−paiwise sensitivity (µ−sensitivity) for TDS. This notion was characterized for ergodic TDS by Huang-Li-Ye in [13]. It was shown that an ergodic TDS is either µ−pairwise sensitive or µ−equicontinutous; (X, T ) is µ−equicontinuous if for every ε > 0 there exists a compact set M such that µ(M ) > 1 − ε and T M is equicontinuous. They showed that every ergodic µ−equicontinuous TDS has discrete spectrum.
We show that if (X, µ) satisfies Lebesgue's density theorem and Vitali's covering theorem (for example when X ⊂ R d or when X is a Cantor space) then (X, T ) is µ−equicontinuous if and only if almost every point is a µ−equicontinuity point (Theorem 9). As a consequence we get that if µ is ergodic then (X, T ) is µ−pairwise sensitive if and only if there are no µ−equicontinuity points (Theorem 29). We also define a subclass of µ−equicontinuous systems that rely on a local periodicity notion and we prove they have discrete rational spectrum when µ is ergodic.
In Section 2 we define and study topological and measure theoretical forms of equicontinuity and local periodicity (µ−equicontinuity and µ − LP ). To characterize µ−equicontinuity we study µ−continuity, which is a property of functions between metric measure spaces (not a dynamical concept). In Section 3, we discuss µ−equicontinuity with respect to invariant measures; we show ergodic µ− LP TDS on Cantor spaces have rational spectrum and we characterize µ−sensitivity.
The characterization of µ−continuity (Theorem 16 ) also helps characterize other weakerer forms of measure theoretical equicontinuity. In [9], µ−mean equicontinuity is introduced; Theorem 16 is applied to characterize these systems, and it is shown that ergodic TDS are µ−mean equicontinuous if and only if they have discrete spectrum.

Topological dynamical systems and measures
Throughout this paper X will denote a compact metric space with metric d. For x ∈ X we represent the balls centered in x with B n (x): = {z | d(x, z) ≤ 1/n}. A topological dynamical system (TDS) is a pair (X, T ) where X is a compact metric space and T : X → X is a continuous transformation.
Two TDS (X 1 , T 2 ), (X 2 , T 2 ) are conjugate (topologically) if there exists a homeomorphism f : An important kind of TDS are TDS on Cantor spaces. The n−window, W n ⊂ Z + is defined as the interval of radius n centred at the origin; a window is an n−window for some n. Let A be a finite set. For W ⊂ Z + and x ∈ X, x W ∈ A |W | represents the restriction of x to W. The set A Z+ endowed with the Cantor metric ( given by d(x, y) = 1 2 m , where m is the largest integer such that x Wm = y Wm ) is called a Cantor space. It is easy to check that Cantor spaces are zero dimensional, i.e. there exists a countable base of clopen sets.
Another important class of systems are Euclidean TDS. We endow the set R d with the Euclidean metric. We say (X, T ) is Euclidean if X ⊂ R d .

Equicontinuity
Mathematical definitions of chaos have been widely studied. Many of them require the system to be sensitive to initial conditions. A TDS (X, T ) is sensitive (or sensitive to initial conditions) if there exists ε > 0 such that for every open set A ⊂ X there exist x, y ∈ A and i ∈ Z + such that d(T i x, T i y) > ε. On the other hand, equicontinuity represents predictable behaviour. As we mentioned a TDS is equicontinuous if T i is an equicontinuous family, and minimal TDS is either sensitive or equicontinuous [1].
We can also define equicontinuity by defining equicontinuity points.
Definition 1 Let (X, T ) be a TDS. We define the orbit metric d o on X as d o (x, y) := sup i≥0 d(T i x, T i y) , and the orbit balls as A TDS (X, T ) is equicontinuous if and only if every x ∈ X is an equicontinuity point. A TDS is equicontinuous if and only if it is uniformly equicontinuous, i.e. for all m ∈ N there exists n ∈ N such that B n (x) ⊂ B o m (x) for every x ∈ X.

µ−Equicontinuity
We will use µ to denote Borel probability measures on X (µ does not need to be invariant under T ). Basedo on [11] we define µ−equicontinuity points.
If x is an equicontinuity point in the support of µ then x is a µ−equicontinuity point.
The following definition appeared in [13].
If (X, T ) is an equicontinuous TDS and µ a Borel probability measure then (X, T ) is µ−equicontinuous. In [8] it was shown that there exists a sensitive TDS on {0, 1} Z such that (X, T ) is µ−equicontinuous for every ergodic Markov chain µ.

Definition 5
We say (X, µ) satisfies Lebesgue's density theorem if for every Borel set A we have that The original Lebesgue's density theorem applies to R d and the Lebesgue measure. If X is a separable metric space that satisfies Besicovitch's covering theorem and µ any Borel probability measure then (X, µ) satisfies Lebesgue's density theorem [14]. In particular this means that if X ⊂ R d and µ is a Borel probability measure then (X, µ) satisfies Lebesgue's density theorem.
Theorem 6 (Levy's zero-one law [6]) Let Σ be a sigma-algebra on a set Ω and P a probability measure.
Corollary 7 Let X be a Cantor space and µ a Borel probability measure. We have that (X, µ) satisfies Lebesgue's density theorem.
Proof. Let m ∈ N, and F n the smallest sigma-algebra that contains all the balls {B n ′ (x)} with n ′ ≤ n . It is easy to see that F ∞ is the Borel sigma-algebra on X. The desired result is a direct application from Theorem 6.
There exist Borel probability spaces that do not satisfy Lebesgue's density theorem [14].
If X ⊂ R d and µ is a Borel probability measure then (X, µ) is Vitali ( [12] pg.8). This result is known as Vitali's covering theorem.
Using a clopen base it is not hard to see that if X is a Cantor space and µ a Borel probability measure then (X, µ) satisfies Vitali's covering theorem. For more conditions see [12].
The following theorem will be proved at the end of the following subsection.

µ−Continuity
Definition 10 Given a TDS (X, T ) we denote the change of metric projection map as f T : .
The function f −1 T is always continuous. A point x ∈ X is an equicontinuity point of (X, T ) if and only if it is a continuity point of f T . Thus (X, T ) is equicontinuous if and only if f T is continuous.

Definition 11
Let X, Y be metric spaces and µ a Borel probability measure on We will define a measure theoretic notion of continuity point (µ−continuity point) that satifies the following property: x ∈ X is a µ−equicontinuity point of T if and only if x is a µ−continuity point of f T . We will show that if (X, µ) is Vitali and almost every x ∈ X is a µ−continuity point of f then f is µ−Lusin; we show the converse is true if (X, µ) satisfies Lebesgue's density theorem (Theorem 16).
In this subsection X will denote a compact metric space, µ a Borel probability measure on X, and Y a metric space with metric d Y (and balls B Y m (y): = {z ∈ Y : d Y (y, z) ≤ 1/m}).

Definition 12
In some cases we can talk about the measure of not necessarily measurable sets. Let A ⊂ X. We say A has full measure if A contains a measurable subset with measure one. We say µ(A) < ε, if there exists a measurable set A ′ ⊃ A such that µ(A ′ ) < ε.
If µ−almost every x ∈ X is a µ−continuity point we say f is µ−continuous.
Every µ−Lusin function is µ−measurable. Lusin's theorem states the converse is true if Y is separable (note that (X, d 0 ) is not necessarily separable). This fact is generalized in the following theorem.
Proof. Let ε > 0. For every µ−equicontinuity point x and m ∈ N there exists N x such that Since ∪ i∈N A i contains the set of µ−continuity points, there exists N such that µ(X A N ) < ε. Since (X, µ) is Vitali there exists a finite set of points F m,ε ⊂ A N and a function n(x) ≥ N such that B n(x) (x) x∈Fm,ε are disjoint and Theorem 16 Let f : X → Y . Consider the following properties If (X, µ) satisfies Lebesgue's density theorem then 1) =⇒ 2).

Proof. 2) + V itali =⇒ 1)
Let m ∈ N and ε > 0. By Lemma 15 there exists a finite set of points F m,ε/2 m and a function n(x) such that Let m ∈ N. Since f | M is continuous then for sufficiently large n and almost every This implies the set of µ−continuity points has measure larger than 1 − ε. We conclude the desired result. Proof of Theorem 9. Take 3) ⇒ 1) If f 0 is µ−Lusin it means that for every κ > 0 there exists a compact set M κ ⊂ X such that µ(M κ ) > 1 − κ and f 0 | Mκ is continuous. This implies that X ′ = ∪ n∈N M 1/n satisfies µ(X ′ ) = 1 and (X ′ , d 0 ) is separable.
For the other implications apply Theorem 16. Note that orbit balls are countable intersections of balls, hence measurable.
The concept of approximate continuity for metric measure spaces has a similar flavour to µ−continuity. It also emulates locally the definition of continuity using measures. The definition of an approximate continuity point is stronger than (2) (in Definition 13) but it does not assume the measurability of any set. A classical result states that if the image of a function is separable then approximate continuity and measurability are equivalent (Theorem 2.9.13 [7]).

Local periodicity
In this subsection we study local periodicity and µ − LEP systems, a subclass of µ−equicontinuous systems.  +(1, 0, ...) (with carrying) map defined on the compact set D = i≥0 Z si (for a survey on odometers see [5]).
Odometers are locally periodic, and that local periodicity is an invariant property under conjugacy.
Subshifts are a special kind of TDS on symbolic spaces where T is the left shift, σ (see [16] for definitions). It is not hard to see that if (X, σ) is a subshift then x ∈ LP (σ) if and only if x is right-periodic.
A closely related concept is regular recurrence. A point x ∈ X is regularly recurrent if for every m there exists p > 0 such that d(x, T ip x) ≤ 1/m for every i ∈ N. Every LP point is regularly recurrent, but regularly recurrent points are not necessarily even LEP (for example Toeplitz subshifts [5]). LEP points are not necessarily regularly recurrent (the point 1000... in a one-sided subshift). If every point of a minimal T DS is regularly recurrent then it is conjugate to an odometer [2], and hence LP.
Equicontinuity and local periodicity are related concepts. Using results from [3] it is easy to show that if X is the unit interval and (X, T ) an equicontinuous TDS then (X, T ) is LEP. Using results from [18] it is easy to see that if X is the unit circle and (X, T ) an equicontinuous TDS then (X, T ) is either an irrational rotation or LEP.
Suppose X a subshift and T : X → X a continuous shift-commuting transformation (these TDS are known as cellular automata or shift endomorphisms). In [8] it is shown that if (X, T ) is LP then it is equicontinuous, that if (X, T ) is equicontinuous then it is LEP, and that there exist non-equicontinuous LEP systems.
Definition 20 Let (X, T ) be a TDS and µ a Borel probability measure on X.
Proof. Let Since (X, T ) is µ − LEP we have that µ(Y ) = 1. Monotonicity of the measure gives the desired result.
Definition 22 Let (X, T ) be a µ − LEP TDS, m ∈ N, and ε > 0. We will use p m ε and pp m ε to denote a particular choice of integers that satisfy the conditions of Lemma 21 and that satisfy that p m ε → ∞ and pp m ε → ∞, as ε → 0. We define Y m ε as in equation (3).
Proof. Let ε > 0 and M := We can express i = pp m ε + j · p + k with j ∈ N and k ≤ p. Using the fact that x, y ∈ Y m ε and pp m ε + k ≤ (p m ε ) 2 + pp m ε we obtain The converse of this proposition is not true in general. An irrational rotation on the circle is equicontinuous, and hence µ−equicontinuous, but contains no LEP points so it is not µ − LEP. For cellular automata there are conditions (for example if X is a shift of finite type and µ a σ−invariant measure) under which µ−equicontinuity implies µ − LEP [8].

Invariant measures and spectral properties
In this section we are interested in measure preserving topological dynamical systems.
We say (M, T, µ) is a measure preserving transformation (MPT) if (M, µ) is a probability measure space, and T : M → M is a measure preserving transformation (measurable and T µ = µ). We say (M, T, µ) is ergodic if it is a MPT and every invariant set has measure 0 or 1.
Two measure preserving transformations (M 1 , T 1 , µ 1 ) and (M 2 , T 2 , µ 2 ) are isomorphic (measurably) if there exists a measure preserving function f : (X 1 , µ 1 ) → (X 2 , µ 2 ), that is 1-1 on a set of full measure, and that satisfies The spectral theory of ergodic systems is particularly useful for studying rigid transformations. We will give the definitions and state the most important results. For more details and proofs see [19].
A measure preserving transformation T on a probability measure space (M, µ) generates a unitary linear operator on the Hilbert space L 2 (M, µ), by U T : f → f • T, known as the Koopman operator. The spectrum of the Koopman operator is called the spectrum of the measure preserving transformation. We say the spectrum of (M, T, µ) is discrete (or pure point) if there exists an orthonormal basis for L 2 (M, µ) which consists of eigenfunctions of the Koopman operator. The spectrum is rational if the eigenvalues are complex roots of unity. Classical results by Halmos and Von Neumann state that two ergodic MPT with discrete spectrum have the same group of eigenvalues if and only if they are isomorphic, and that an ergodic MPT has discrete spectrum if and only if it is isomorphic to a rotation on a compact metric group. The eigenfunctions of a rotation on a compact group are generated by the characters of the group.
The converse is not true. For example Sturmian subshifts with their unique invariant measure have discrete spectrum and it is not hard to see they have no µ−equicontinuity points. Nonetheless using a weaker version of µ−equicontinuity (µ−mean equicontinuity) it is possible to characterize discrete spectrum for ergodic TDS (see [9]).
In the next subsection we show that ergodic µ − LEP TDS on Cantor spaces have rational spectrum. Proof. Since (X, T ) is µ−equicontinuous we have there exists X ′ ⊂ X such that X ′ is d o −separable and µ(X ′ ) = 1 (Theorem 9). If x ∈ LEP (T ) LP (T ) then by Lemma 25 there exists m such that T n µ(B o m (x)) → 0. Using the fact that µ is an invariant measure we get that µ(B o m (x)) = T n µ(B o m (x)) = 0. This implies µ(X ′ ∩ LEP (T ) LP (T )) = 0. Therefore µ(LP (T )) = 1.

Local periodicity
Theorem 27 Let X be a Cantor space, (X, T ) a TDS and µ an invariant measure. If (X, T ) is µ − LEP then (X, T, µ) has discrete rational spectrum.
Proof. Using Proposition 26 we have that (X, T ) is µ − LP.
Using the fact that y ∈ LP (T ) and X is a Cantor space (see Remark 18) we have that B o m (T pm(y) y) = B o m (y) and This means 1 B o m (y) ∈ Span {f m,y,k } k . Let n ∈ N, and x ∈ X. Since (X, T ) is µ− equicontinuous there exist sequences {y i } ⊂ LP (T ) and {m i } ⊂ N, such that B o mi (y i ) are disjoint and µ(B n (x)) = µ(∪B o mi (y i )) (use Theorem 9 bullet 3) ). This means 1 Bn(x) can be approximated in L 2 by elements in Span 1 B o m (y) : m ∈ N and y ∈ LP (T ) . Since the closure of Span 1 Bn(x) : n ∈ N, x ∈ X is L 2 (X, µ) we conclude the closure of Span {f m,y,k } m,y,k is L 2 (X, µ).
This result is particularly useful for cellular automata, as µ − LEP invariant measures appear naturally as the limit measures of µ−equicontinuous CA; for more information see [8].

µ−Sensitivity
The following notion of measure theoretical sensitivity was introduced in [4], and it was characterized in [13]. We provide another characterization.
In [13] it was shown that if µ is ergodic then either (X, T ) is µ−sensitive or µ−equicontinuous. We obtain another characterization of µ−sensitivity.
Theorem 29 Let (X, µ) be a Vitali space that satisfies Lebesgue's density theorem (X, T ) be a TDS and µ an ergodic measure. Then (X, T ) is µ−sensitive if and only if X contains no µ−equicontinuity points.
Proof. Let E µ be the set of µ−equicontinuity points.
By Theorem 9 and the previous comment we have that T is µ−sensitive or E µ has measure one.