Category of Attractor and Its Application

In this paper, we provide a new approach to study the geometry of attractor. By applying category, we investigate the relationship between attractor and its attraction basin. In a complete metric space, we prove that the categories of attractor and its attraction basin are always equal. Then we apply this result to both autonomous and non-autonomous systems, and obtain a number of corresponding results.


Introduction
Attractors of a given system are of crucial importance, this is because that much of longtime dynamics is represented by the dynamics on and near the attractors.It is well known that the global attractors of dynamical systems can be very complicated.The geometry can be very pathological, even in the finite dimensional situation.To have a better understanding on the dynamics of a system, it is quite necessary for us to study the topology and geometry of the attractors.In the past few decades, there appeared many studies.In [1], Kapitanski and Rodnianski studied the shape of attractors of continuous semi-dynamical systems on general metric spaces.They proved that the global attractor has the same shape as the state space.Moreover, using the results on the shape of attractors, they developed an elementary Morse theory for an attractor.Lately, the author of [2] studied the Morse theory of attractors for semiflows on complete metric spaces by constructing continuous Lyapunov functions, and he introduced the concept of critical groups for Morse sets and established Morse inequalities and Morse equations for attractors.To study the geometry of the attractors, some concepts such as Lyapunov exponents, the Hausdorff dimension and the fractal dimension were also proposed, see [3] [4] etc. Recently, in [5] author studied the geometrical property of the global attractor for a class of symmetric p-Laplacian equations by means of 2 Z index, obtained some lower estimates for the fractal dimension of the global attractor.In this paper, by using Ljusternik-Schnirelmann category (category for short), we try to provide a new approach to studying the geometry of the global attractor.Category is a topological invariant, which often be used in the estimate of the lower bound of the number of critical points, see [6].Here we investigate the relationship between attractor and attraction basin in the sense of category.In a complete metric space, for asymptotic compact semiflow, we obtain that the categories of attractor and attraction basin are always equal.This result match with the result in [1].Now we can directly describe this result by category.The result will be of most interest when we choose X be some special metric space.Finally, we have to point out that it is generally not very easy to compute the category of a given space.However, we can see there are more and more new results and methods about calculation of category, see [7] [8] etc.
We will prove the main results in Section 3 and give some applications in Section 4. Before that we provide some preliminaries and results in Section 2.

Preliminaries
We recall some basic definitions and facts in the theory of dynamical systems for semiflows on complete metric spaces.Let X be a complete metric space with metric ( , ) d ⋅ ⋅ .Definition 2.1 A semiflow (semidynamical system) on X is a continuous mapping : From now on, we will always assume that there has been given a semidynamical system ( ) S t on X ; Moreover, we assume ( ) S t is asymptotically compact, that is, ( ) S t satisfies the following assumption: ( ) : A For any bounded sequence n x X ∈ and n t → +∞ , if the sequence ( ) S t x is bounded, then it has a convergent subsequence.
The asymptotic compactness property (A) is fulfilled by a large number of infinite dimensional semiflows generated by PDEs in application [4].
Let A be a subset of X .We say that A attracts B X ⊂ , if for any 0 ε > there exists a 0 The attraction basin of A , denote by ( ) , is defined as: The set A is said to be positively invariant (resp.invariant), if ( ) ( .( ) ), 0.

S t A A resp S t A
is said to be an attractor of ( ) S t , if it is invariant and attracts a neighborhood of U itself.An attractor  is said to be the global attractor of ( ) S t , if it attracts each bounded subset of X .
Let U be an open subset of X , and K be a closed subset of In order to prove our result, we need following theorem (see Theorem 3.5 in [2]).Let there be given an attractor  with attraction basin ( ) 5 We emphasize that the ( ) V x is coercive with  on Ω .This point is not contained in the statement of Theorem 2.4, but we can obtain this result from the proof of the Theorem 3.5 in [2] easily.
In the following, we recall some basic results on the Ljusternik-Schnirelmann category (category for short).Definition 2.6 Let M be a topological space, A M ⊂ be a closed subset.Set The category defined above has properties as follows.Lemma 2.7 Properties for the category: 1) ( ) 0 For the proof of this lemma, we refer readers to [6].Remark 2.8 By (2) (5), we can easily obtain that if A is compact, then there exists a ε -neighborhood ( , ) Just by the definition of category, we can prove the following lemma: Lemma 2.9 Let , X Σ are topology spaces, and

Category of Attractor
The main results can be stated as follows: Theorem 3.1 Let X be a complete metric space and ( ) S t is a semiflow on X , which is asymptotically compact.Let  be an attractor of ( ) S t on X with attraction basin ( ) Since  is compact, by continuity (Remark 2.8}), fixed 0 ε > small enough, we have If we find a set ( , ) by using monotonicity again and (3.2}), we have ( ) ( ). cat cat Then combine (3.1}) and (3.4), we will obtain the result ( ) ( ). cat cat Ω Ω Ω =  Now the rest of the work in this proof is in finding the appropriate set K , which is subset of ( , ) B ε  and sa- tisfies (3.3).In order to obtain the proper set K , the key tool here is the level set of Lyapunov function on attractor  .Thanks to Theorem 2.4, we can construct a Lapunov function ( ) By the Remark 2.5, ( ) V x is coercive with  , that is for the fixed ε above, there exists 0 δ > such that ( ) 0, \ ( , ).
⊂  .We use the method in [2], Define a function ( ) Here ( ) t x < +∞ and ( ) t x is continuous on Ω .(See Theorem 5.1 and Lemma 5.2 in [2], in which Ω replaced by b t x is continuous on Ω , we see that η is a continuous mapping, by deformation nondecreasing and monotonicity, we have Now we just let a K V = , which completes the proof.Now to extend our result to non-autonomous case, we consider a skew-product system, which consists of a base semiflow, and a semiflow on the phase space that is in some sense driven by the base semiflow.More precisely, the base semiflow consists of the base space Σ , which we take to be a metric space with metric ρ , and a group of continuous transformations { } The dynamics on the phase space ( , ) X d is given by a family of continuous mappings satisfy the cocycle property 1) (0, )

Applications
In this section, we further apply our results to some special metric space X , we will see some interesting results.

Example 1. Assume ( )
n n X S or T = .Let ( ) S t is a asymptotically compact semiflow on X .If  is a global attractor of ( ) S t on X .Then X =  .Proof.Suppose the contrary.Then there exist at least one point s X ∈ such that s ∉  .Then we deduce that \{ } X s X ⊂ ⊂  . By the monotonicity, we have ( \{ }) .
Thus, we have .
X X cat cat X <  On the other hand, by virtue of Theorem, we have , which leads to a contradiction!Hence, the global attractor  must be phase space X itself.
Using similar arguments, one can prove the case of n X T = .
Example 2. In skew-product flow case, we assume φ θ , where σ π  is the section of  over σ ∈ Σ .Since corollary 3.2, we can show that the pull back attractor of the skew-product flow ( , ) φ θ must be ( ) ) [9]a asymptotically compact semiflow on  .If  is a global attractor of ( ) T t on  .Then =  .Proof.Suppose the contrary.Then there exist at least one point s X ∈ such that s ∉  .By Theorem 15.7 in[9], if  is a global attractor of ( ) S t on  .Then ( ) A ⋅ with ( ) A  is the pullback attractor of the skew-product flow ( , )