On the Residual Continuity of Global Attractors

Abstract

In this brief paper, we studied the residual continuity of global attractors ${\mathcal{A}}_{\lambda }$ in varying parameters $\lambda \in \Lambda$ with $\Lambda$ a bounded Borel set in ${\mathbb{R}}^d$. We first reviewed the well-known residual continuity result of global attractors and then showed that this residual continuity is equivalent to the dense continuity. Then, we proved an analogue continuity result in measure sense that, under certain conditions, the set-valued map $\lambda \mapsto {\mathcal{A}}_{\lambda }$ is almost (in the Lebesgue measure sense) uniformly continuous: for any small $\epsilon >0$ there exists a closed subset $C_{\epsilon }\subset \Lambda$ with Lebesgue measure $m\left(C_{\epsilon }\right)>\mu (\Lambda )-\epsilon$ such that the set-valued map $\epsilon \mapsto {\mathcal{A}}_{\epsilon }$ is uniformly continuous on $C_{\epsilon }$. This, in return, indicates that the selected attractors $\{{\mathcal{A}}_{\lambda }:\lambda \in C_{\epsilon }\}$ can be equi-attracting.

1. Introduction

The global attractor of a dynamical system is a compact invariant set that attracts the trajectories starting in any bounded set in the phase space. It exists widely for finite- and infinite-dimensional dissipative dynamical systems, and once the attractor exists, it is often finite dimensional [1]. Hence, a number of monographs on the attractor theory can be found in the literature; see, for instance, Babin and Vishik [2], Hale [3], Ladyzhenskaya [4], Robinson [5] and Temam [1], etc.

If a dynamical system $S_{\lambda }$ on a Banach space X has a parameter $\lambda$, where $\lambda$ comes from a metric space $\Lambda$, it is natural to inquire if the attractor ${\mathcal{A}}_{\lambda }$ is robust as $\lambda$ varies. We define a metric space $\mathcal{E}$ with $\mathcal{E}=\{E:E\mathrm{is}\mathrm{a}\mathrm{compact}\mathrm{set}\mathrm{in}X\}$ endowed with the metric

$${\varrho }_{\mathcal{E}}(A,B):=max\{dist(A,B),dist(B,A)\},$$

where $dist(A,B)$ denotes the Hausdorff semimetric between subsets of X. Then, the robustness of the attractor ${\mathcal{A}}_{\lambda }$ reduces to the continuity of the set-valued map ${\mathcal{A}}_{\lambda }:\Lambda \to \mathcal{E},$$\lambda \mapsto {\mathcal{A}}_{\lambda }$. Babin and Pilyngin [6], and more recently Hoang, Olson and Robinson [7,8], showed that under certain conditions ${\mathcal{A}}_{\lambda }$ can have a residual continuity: the points at which ${\mathcal{A}}_{\lambda }$ is continuous form a residual subset of $\Lambda$, where by a residual set we name a set whose complement is of the first category (of Baire, i.e., if the complement can be written as a countable union of nowhere dense sets). The residual continuity gives an impression that ${\mathcal{A}}_{\lambda }$ is continuous at a very large set of points in $\Lambda$, so it has been applied to describe the robustness and approximation of attractors; see, e.g., [6,9,10,11,12,13].

However, Oxtoby [14] (Theorem 1.6) showed that a residual set in $\mathbb{R}$ can have a Lebesgue measure of zero, so the set of continuity points of ${\mathcal{A}}_{\lambda }$, though residual, can be a null set. In other words, a large set in the residual set sense can be a small set in the measure sense. Hence, it is necessary to discuss the measure of the set of these continuity points.

In this paper, we first review the Baire category theorem and prove that the set of the continuity points of a function is residual if and only if it is dense; see Theorem 2. Then, for $\Lambda$, a bounded Borel set in ${\mathbb{R}}^d$, we study the almost (in the Lebesgue measure sense) uniform continuity: for any small $\epsilon >0$ there exists a closed subset $C_{\epsilon }\subset \Lambda$ with Lebesgue measure $m\left(C_{\epsilon }\right)>\mu (\Lambda )-\epsilon$ such that the set-valued map $\epsilon \mapsto {\mathcal{A}}_{\epsilon }$ is uniformly continuous on $C_{\epsilon }$ and the selected attractors $\{{\mathcal{A}}_{\lambda }:\lambda \in C_{\epsilon }\}$ can be equi-attracting; see Theorem 4.

2. Residual Continuity & Dense Continuity

Let $\Lambda$ be a complete metric space. A subset $A\subset \Lambda$ is called nowhere dense if the interior of the closure of A is empty, i.e., ${\left(\overline{A}\right)}^{\circ }=\varnothing$. The nowhere dense set $A=\{1/n:n\in \mathbb{N}\}$ in $\Lambda =\mathbb{R}$ indicates that a nowhere dense set is not necessarily closed, so its complement is not necessarily open. Nevertheless, the following useful lemma holds.

Lemma 1.

A set $A\subset \Lambda$ is nowhere dense if and only if its complement $A^c$ contains an open dense set in Λ.

Proof. 

Since A is nowhere dense, then so is its closure $\overline{A}$. Then the complement ${\left(\overline{A}\right)}^c$ of $\overline{A}$ is open and is dense in $\Lambda$ by Baire’s theorem. Therefore, the necessity holds. The reverse is elementary to prove. □

A set A in $\Lambda$ is called of first category (or meager) if it can be written as a countable union of nowhere dense sets. The complement of a meager set is called residual. For example, in $\Lambda =\mathbb{R}$, every finite set is nowhere dense and of first category (in fact, this is true for all metric spaces which has no isolated points); the set $\mathbb{Q}$ of rationals and the set $\overline{\mathbb{Q}}$ of irrationals are both dense, while $\mathbb{Q}$ is of first category and $\overline{\mathbb{Q}}$ is residual. This gives the intuition that a residual set “means more” than merely the density. In fact, the following Baire category theorem shows that any residual set in a complete metric space is dense, while the above example of $\mathbb{Q}$ is a counterexample for the reverse.

Theorem 1.

(Baire category theorem) Let Λ be a complete metric space. Then

(i)

a set of first category has an empty interior;

(ii)

a residual set is dense;

(iii)

a countable intersection of dense open sets is dense.

However, we have to note that [14] (Theorem 1.6) gives example subsets of $\Lambda \subset \mathbb{R}$ which are of first category but have full Lebesgue measure. This indicates that the sets of first category and the sets of a measure of zero are "small" only in their own sense, and neither of the two implies the other.

For the latter purpose, we note the following lemma.

Lemma 2.

Any $F_{\sigma }$ set D in a complete metric space Λ is meager if and only if its complement $D^c$ is dense.

Proof. 

Suppose that $D={\cup }_{i\in \mathbb{N}}{D}_i$, with each $D_i$ a closed set in $\Lambda$. If D is meager, then $D^c$ is residual and thereby dense by Baire category theorem. Conversely, if $D^c$ is dense, then every $D_i^c\supset D^c$ is dense and open (as $D_i$ is closed), so $D_i$ is equivalently nowhere dense, and therefore, D is meager. □

Let $f:\Lambda \to \mathcal{E}$ be any function between two complete metric spaces $\Lambda$ and $\mathcal{E}$. We denote by D the set of points in $\Lambda$ on which f is discontinuous, and by $D^c$ the complement, or the set of continuity points of f. f is called residually (resp. densely) continuous if $D^c$ is residual (resp. dense). The following theorem shows that the residual continuity of f does not mean more than the dense continuity; they are in fact equivalent.

Theorem 2

(Residual continuity ⇔ dense continuity). Let $f:\Lambda \to \mathcal{E}$ be a function between complete metric spaces Λ and $\mathcal{E}$. Then, the set $D^c$ of continuity points of f is residual if and only if it is dense in Λ.

Proof. 

Since $\Lambda$ is complete, the necessity follows directly from the Baire category theorem.

To see the sufficiency, we suppose that $D^c$ is dense. Define the oscillation of f on $\Lambda$ by

$$\omega (\Lambda )=\underset{x,y\in \Lambda }{sup}\varrho (f\left(x\right),f\left(y\right)).$$

For any fixed $x\in \Lambda$, the value

$$\omega \left(x\right)=\underset{\sigma \to 0^{+}}{lim\; sup}\varrho (f(x-\delta ),f(x+\delta ))$$

is called the oscillation of f at x. $\omega \left(x\right)$ is a quantitative measure of the size of the discontinuity of f at x, and $\omega \left(x\right)=0$ if f is continuous at x. Clearly, the set of discontinuity points can be represented as

$$\begin{array}{c}\hfill D=\bigcup _{n\in \mathbb{N}}\left\{x\in \Lambda :\omega \left(x\right)⩾\frac{1}{n}\right\}.\end{array}$$

(1)

Notice that each set $\{x\in \Lambda :\omega (x)<\epsilon \}$ is open, as there is apparently a neighborhood of x such that $\omega \left(x^{\prime }\right)<\epsilon$ for all $x^{\prime }$ in the neighborhood, so D is an $F_{\sigma }$ set (countable union of closed sets). Now to prove that $D^c$ is residual, or, equivalently, D is meager, we only need to prove that every $F_{\sigma }$ set with dense complement is meager, which is concluded in Lemma 2. □

The following lemma shows that the pointwise limit of continuous functions on compact metric spaces is at least residual continuous.

Lemma 3

([15]). Let I be a compact metric space and Y a metric space. Suppose that $f_n:I\to Y$, $n\in \mathbb{N}$, is a family of continuous map. If $f_n$ converges pointwise to f, i.e., $f_n\left(\epsilon \right)\to f\left(\epsilon \right)$ for each $\epsilon \in I$, then the points of continuity of f form a residual subset of I.

3. Almost Uniform Continuity in Lebesgue Measure Sense

The residual continuity means that the set of continuity points of a function is residual. However, a residual set in, for example, $\Lambda =\mathbb{R}$ can have a Lebesgue measure of zero; see [14]. In this section, we study an almost (in the Lebesgue measure sense) uniform continuity.

We denote by ${\mathbb{R}}^d$ the d-dimensional Euclidean space, and by $\mathcal{B}\left({\mathbb{R}}^d\right)$ the Borel $\sigma$-algebra where the Lebesgue measure m is defined. A set in $\mathcal{B}\left({\mathbb{R}}^d\right)$ is called a Borel set. A function f defined on a bounded Borel set $\Lambda \in \mathcal{B}\left({\mathbb{R}}^d\right)$ and mapping to a metric space $\mathcal{E}$ is called almost uniformly continuous on $\Lambda$ if for any small $\epsilon >0$ there exists a closed set $C_{\epsilon }\subset \Lambda$ with a Lebesgue measure $m\left(C_{\epsilon }\right)>m(\Lambda )-\epsilon$ such that the restriction ${f|}_{C_{\epsilon }}$ of f to $C_{\epsilon }$ is uniformly continuous.

Since $\Lambda$ is bounded, without loss of generality, we assume that its Lebesgue measure $m(\Lambda )=1$. Then, we have a probability space $(\Lambda ,\mathcal{F},m)$, where $\mathcal{F}$ denotes the Borel $\sigma$-algebra on $\Lambda$. The following regularity result of Borel probability measures is crucial for us; see, for instance, [15] (Theorem 6.1).

Lemma 4.

Let X be a metric space with Borel σ-algebra $\mathcal{B}\left(X\right)$. Then, any probability measure $\mathbb{P}$ on $(X,\mathcal{B}(X\left)\right)$ is regular: for any $B\in \mathcal{B}\left(X\right)$ and $\epsilon >0$ there exists an open set $U_{\epsilon }$ and a closed set $C_{\epsilon }$ with $C_{\epsilon }\subset B\subset U_{\epsilon }$ and $\mathbb{P}(U_{\epsilon }\backslash C_{\epsilon })<\epsilon$.

Lemma 5

(Egroff’s theorem). If a sequence of measurable functions $g_n$ defined on a measure space $(\Omega ,\mathcal{F},m)$ converges to g pointwise on a set E of finite measure, then for each $\epsilon >0$, there is a set $F\subset E$ with $m\left(F\right)<\epsilon$ such that $g_n$ converges to g uniformly on $E\backslash F$.

Lemma 6.

Suppose that $(X,{\varrho }_X)$ and $(Y,{\varrho }_Y)$ are two metric spaces (which need not be complete). If a sequence of uniformly continuous functions $g_n:X\to Y$ uniformly converges to a function $g:X\to Y$, then the limit function g is uniformly continuous on X.

Proof. 

For any $\epsilon >0$, since the sequence ${\left\{g_n\right\}}_{n\in \mathbb{N}}$ uniformly converges to g, there is an $n_{*}\in \mathbb{N}$ such that

$$\underset{x\in X}{sup}{\varrho }_Y(g_{n_{*}}\left(x\right),g\left(x\right))<\epsilon /3.$$

In addition, since the function $g_{n_{*}}$ is uniformly continuous on X, there is a $\delta =\delta \left(\epsilon \right)>0$ such that for all $x_1,x_2\in X$ with ${\varrho }_X(x_1,x_2)<\delta$, we have

$${\varrho }_Y(g_{n_{*}}\left(x_1\right),g_{n_{*}}\left(x_2\right))<\epsilon /3.$$

Therefore, for all such $x_1,x_2$ with ${\varrho }_X(x_1,x_2)<\delta$,

$$\begin{array}{cc}\hfill {\varrho }_Y(g\left(x_1\right),g\left(x_2\right))& ⩽{\varrho }_Y(g\left(x_1\right),g_{n_{*}}\left(x_1\right))+{\varrho }_Y(g_{n_{*}}\left(x_1\right),g_{n_{*}}\left(x_2\right))+{\varrho }_Y(g_{n_{*}}\left(x_2\right),g\left(x_2\right))\hfill \\ & ⩽2\underset{x\in X}{sup}{\varrho }_Y(g_{n_{*}}\left(x\right),g\left(x\right))+{\varrho }_Y(g_{n_{*}}\left(x_1\right),g_{n_{*}}\left(x_2\right))\hfill \\ & <2\epsilon /3+\epsilon /3=\epsilon ,\hfill \end{array}$$

so g is uniformly continuous on X. □

The following theorem shows that the limit of continuous functions on bounded Borel sets is almost uniformly continuous.

Theorem 3.

Suppose that ${\left\{f_n\right\}}_{n\in \mathbb{N}}$ is a sequence of continuous functions from a bounded Borel set $\Lambda \subset {\mathbb{R}}^d$ to a metric space $\mathcal{E}$. If there is a function $f:\Lambda \to \mathcal{E}$ such that $f_n\to f$ pointwise on Λ (i.e., $f_n\left(\lambda \right)\to f\left(\lambda \right),\forall \lambda \in \Lambda$), then for any $\epsilon >0$ there is a closed measurable subset $C_{\epsilon }\subset \Lambda$ with measure $m(\Lambda \backslash C_{\epsilon })<\epsilon$ such that

(i)

$f_n\to f$ uniformly on $C_{\epsilon }$;

(ii)

the restriction ${f|}_{C_{\epsilon }}$ of f to $C_{\epsilon }$ is uniformly continuous.

Proof. 

Since $\Lambda$ is bounded, without loss of generality we assume that $m(\Lambda )=1$. Since $f_n\to f$ pointwise on $\Lambda$, by Egroff’s theorem, for any $\epsilon >0$ there is a set $A_{\epsilon }\subset \Lambda$ with $m\left(A_{\epsilon }\right)<\epsilon /2$ such that $f_n\to f$ uniformly on $B_{\epsilon }:=\Lambda \backslash A_{\epsilon }.$ Clearly, $B_{\epsilon }\in \mathcal{F}$, and $m\left(B_{\epsilon }\right)>1-\epsilon /2$. Then for this Borel set $B_{\epsilon }$, by Lemma 4 there exists a closed set $C_{\epsilon }$ included in $B_{\epsilon }$ such that $m(B_{\epsilon }\backslash C_{\epsilon })<\epsilon /2$ or, equivalently, $m\left(C_{\epsilon }\right)>m\left(B_{\epsilon }\right)-\epsilon /2$. Since $m\left(B_{\epsilon }\right)>1-\epsilon /2$, the closed set $C_{\epsilon }$ is of measure $m\left(C_{\epsilon }\right)>1-\epsilon$. Since $C_{\epsilon }$ is closed and finite-dimensional, by the Heine–Cantor theorem each continuous function $f_n$ on $C_{\epsilon }$ is uniformly continuous on $C_{\epsilon }$. As we have already showed that $f_n\to f$ uniformly on $B_{\epsilon }$ and also on $C_{\epsilon }$, by Lemma 6, the limit function f is uniformly continuous on $C_{\epsilon }$; the theorem is proved. □

4. Residual Continuity and Almost Uniform Continuity of Global Attractors

In this section we apply Theorem 3 to attractor theory to obtain the almost uniform continuity of global attractors. Recall that a semigroup S on a Banach space X is a mapping $S(·,·):[0,\infty )\times X\to X$ such that $S(0,·)$ is the identity on X and $S(t+s,x)=S(t,S(s,x\left)\right)$ for all $t⩾s⩾0$ and $x\in X$. A semigroup is called continuous if it is continuous in x. The global attractor $\mathcal{A}$ of a semigroup S is a compact set in X which is invariant, i.e., $S(t,\mathcal{A})=\mathcal{A}$ for all $t⩾0$, and attracts every bounded set E in X: ${lim}_{t\to \infty }dist(S(t,E),\mathcal{A})=0$, where $dist(A,B)={sup}_{a\in A}{inf}_{b\in B}{\parallel a-b\parallel }_X$ denotes the Hausdorff semi-metric between subsets of X.

Let $\Lambda$ be a metric space and ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in \Lambda }$ a parametrized family of global attractors. ${\mathcal{A}}_{\lambda }$ is called continuous at ${\lambda }_0$ if the set-valued map ${\mathcal{A}}_{\lambda }:\Lambda \to \mathcal{E},\lambda \mapsto {\mathcal{A}}_{\lambda }$ is continuous at ${\lambda }_0$, where

$$\mathcal{E}=\{E:E\mathrm{is}\mathrm{a}\mathrm{compact}\mathrm{set}\mathrm{in}X\},$$

endowed with the metric

$${\varrho }_{\mathcal{E}}(A,B):=max\left\{dist(A,B),dist(B,A)\right\},\forall A,B\in \mathcal{E}.$$

Before our almost uniform continuity theorem, we note the following well-known residual continuity of global attractors; see [6] for $\Lambda$ as a topological space and [7] for $\Lambda$ as a compete metric space. See also [8,13] for an adaption to pullback and uniform attractors and [9,10,11,12] for applications in random attractors and their Wong–Zakai approximation. For the latter purpose, we here restrict ourselves to $\Lambda \subset {\mathbb{R}}^d$.

Lemma 7.

LetΛbe a bounded subset of ${\mathbb{R}}^d$ with finite Lebesgue measure and ${\left\{S_{\lambda }\right\}}_{\lambda \in \Lambda }$ a parametrized family of semigroups on X. Suppose that

(i)

$S_{\lambda }$ has a global attractor ${\mathcal{A}}_{\lambda }$ for every $\lambda \in \Lambda$;

(ii)

There is a bounded subset D of X such that ${\mathcal{A}}_{\lambda }\subset D$ for every $\lambda \in \Lambda$; and

(iii)

For $t>0$, $S_{\lambda }(t,x)$ is continuous in Λ uniformly for x in bounded subsets of X.

Then the points at which the ${\mathcal{A}}_{\lambda }$ is continuous form a residual subset of Λ.

Now under the same conditions as Lemma 7, we apply Theorem 3 to obtain the almost uniform continuity of global attractors.

Theorem 4.

The assumptions are those in Lemma 7, i.e., let Λ be a bounded subset of ${\mathbb{R}}^d$ with finite Lebesgue measure and $S_{\lambda }$ be a parametrized family of continuous semigroups on X, and suppose that

(i)

$S_{\lambda }$ has a global attractor ${\mathcal{A}}_{\lambda }$ for every $\lambda \in \Lambda$;

(ii)

There is a bounded subset D of X such that ${\mathcal{A}}_{\lambda }\subset D$ for every $\lambda \in \Lambda$; and

(iii)

For $t>0$, $S_{\lambda }(t,x)$ is continuous in λ uniformly for x in bounded subsets of X.

Then

(i)

for any $\epsilon >0$, there is a closed measurable subset $C_{\epsilon }\subset \Lambda$ with Lebesgue measure $m\left(C_{\epsilon }\right)>m(\Lambda )-\epsilon$ such that the family of global attractors ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in C_{\epsilon }}$ is uniformly continuous on $C_{\epsilon }$;

(ii)

if D is a uniformly absorbing set (i.e., for any bounded set $E\subset X$, ${\cup }_{\lambda \in \Lambda }{S}_{\lambda }(t,E)\subset D$ for t large enough), then for the closed set $C_{\epsilon }\subset \Lambda$ defined above, the global attractors ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in C_{\epsilon }}$ are equi-attracting: for any bounded set $E\subset X$,

$$\underset{\lambda \in C_{\epsilon }}{sup}dist(S_{\lambda }(t,E),{\mathcal{A}}_{\lambda })\to 0,ast\to \infty .$$

Proof. 

Since D is a common absorbing set, the global attractor has the structure

$${\mathcal{A}}_{\lambda }=\bigcap _{k\in \mathbb{N}}\overline{{\bigcup }_{n=k}^{\infty }{S}_{\lambda }(n,D)},\lambda \in \Lambda ,$$

where

$$S_{\lambda }(n,D)=\bigcup _{x\in D}{S}_{\lambda }(n,x).$$

We write

$$\begin{array}{c}\hfill f_n\left(\lambda \right):=S_{\lambda }(n,D).\end{array}$$

(2)

Then for every n fixed, $\lambda \mapsto f_n\left(\lambda \right)$ is a function from $\Lambda$ to the metric space $\mathcal{E}$ with

$$\mathcal{E}=\{E:E\mathrm{is}\mathrm{a}\mathrm{compact}\mathrm{set}\mathrm{in}X\},$$

endowed with the metric ${\varrho }_{\mathcal{E}}(A,B):=max\{dist(A,B),dist(B,A)\},$ where $dist(A,B)$ denotes the Hausdorff semimetric between subsets of X.

Then the condition (ii) indicates that for every n fixed, $f_n\left(\lambda \right)$ is continuous in $\lambda$. In addition, for any $\lambda \in \Lambda$ fixed, since a point y belongs to ${\mathcal{A}}_{\lambda }$ if there is a sequence $x_n\in D$ such that $S_{\lambda }(n,x_n)\to y$, we know that

$${\varrho }_{\mathcal{E}}(f_n\left(\lambda \right),{\mathcal{A}}_{\lambda })\to 0,\mathrm{as}n\to \infty ,$$

or, in other words, the set-valued mapping $\lambda \mapsto {\mathcal{A}}_{\lambda }$ is the pointwise limit of the mappings $f_n\left(\lambda \right)$. Then from Theorem 3 (ii) we have the conclusion (i).

To see (ii), it suffices to derive from Theorem 3 (i) that $f_n\left(\lambda \right)=S_{\lambda }(n,D)\to {\mathcal{A}}_{\lambda }$ uniformly in $\lambda \in C_{\epsilon }$ and observe that D is a uniformly absorbing set. □

Example 1.

Let us consider the following Lorenz system of three ordinary differential equations

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill x^{\prime }& =-\sigma x+\sigma y,\hfill \\ \hfill y^{\prime }& =rx-y-xz,\hfill \\ \hfill z^{\prime }& =-bz+xy,\hfill \end{array}\right.\end{array}$$

(3)

where σ, r and b are positive constants. As an example, let us consider the continuity of the global attractor of the system with respect to the parameter $\lambda =(\sigma ,r,b)\in \Lambda :={(0,1)}^3$ in ${\mathbb{R}}^3$. As a fundamental finite-dimensional system, for every $\lambda \in {(0,1)}^3$, the Lorenz system (3) generates a continuous semigroup $S_{\lambda }$ which has a global attractor ${\mathcal{A}}_{\lambda }$; see, for instance, Temam [1]. In fact, standard uniform estimates on solutions show that these attractors ${\mathcal{A}}_{\lambda }$ are bounded in a bounded set D, and from studying the difference of two solutions with different parameters, it follows the continuity of $S_{\lambda }$ in λ. In other words, the conditions of Lemma 7 are satisfied, and therefore Hoang et al. [8] deduced that the parameterized global attractor ${\mathcal{A}}_{\lambda }$ is residual continuous in $\lambda \in \Lambda$. Now, by Theorem 4, we know that ${\mathcal{A}}_{\lambda }$ is almost uniform continuous; also, for any $\epsilon >0$ there is a closed measurable subset $C_{\epsilon }\subset \Lambda$ with Lebesgue measure $m\left(C_{\epsilon }\right)>1-\epsilon$ such that ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in C_{\epsilon }}$ is uniformly continuous on $C_{\epsilon }$. In addition, these attractors ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in C_{\epsilon }}$ are equi-attracting.

5. Conclusive Comments

By Theorem 4, we proved that under quite admissible conditions, the family ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in \Lambda }$ of global attractors can be almost uniformly continuous: for any $\epsilon >0$ there is a closed subset $C_{\epsilon }\subset \Lambda$ with Lebesgue measure $m\left(C_{\epsilon }\right)>m(\Lambda )-\epsilon$ such that the family ${\left\{A_{\lambda }\right\}}_{\lambda \in C_{\epsilon }}$ of attractors is uniformly continuous and equi-attracting. This can be regarded as a rough analogue (in the measure sense) for Lemma 7 (in the category sense).

It is worth noting what Theorem 4 does not say: that the family ${\left\{{\mathcal{A}}_{\epsilon }\right\}}_{\epsilon \in C_{\epsilon }}$ is uniformly continuous does not mean that the original family ${\left\{\mathcal{A}\right\}}_{\epsilon \in \Lambda }$ is continuous at $C_{\epsilon }$. In other words, if we denote by $D^c$ the set of continuity points in $\Lambda$ at which the set-valued map $\lambda \mapsto {\mathcal{A}}_{\lambda }$ is continuous, then we do not have $C_{\epsilon }\subset D^c$. In fact, as the complement of an $F_{\sigma }$ set (1), $D^c$ is measurable, so if $C_{\epsilon }\subset D^c$, we would have $m(\Lambda )⩾m\left(D^c\right)⩾m\left(C_{\epsilon }\right)>m(\Lambda )-\epsilon$ for any $\epsilon >0$, indicating that $m\left(D^c\right)=m(\Lambda )$, i.e., ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in \Lambda }$ would then be continuous almost everywhere.

Since Lemma 7 showed that the set D of discontinuity points is of the first category, it is natural to inquire whether or not it is of a measure of zero. As we explained above, Theorem 4 does not give an answer. In fact, even if the set $D^c$ of the continuity points of ${\left\{A_{\lambda }\right\}}_{\lambda \in \Lambda }$ has a positive measure is unknown. Although we constructed a closed set $C_{\epsilon }$ with positive measure, such a set can have an empty interior (i.e., $C_{\epsilon }$ can be a nowhere dense set in $\Lambda$), so every point in $C_{\epsilon }$ can be a discontinuity point of ${\left\{{\mathcal{A}}_{\lambda }\right\}}_{\lambda \in \Lambda }$. Therefore, the almost everywhere continuity of global attractors is not covered in this paper and deserves further study.