1. Introduction and Preliminaries
Topological entropy, which describes the complexity of a system, plays an important role in topological dynamical systems. It was first induced by Adler et al.  as an invariant of topological conjugacy. Later, Bowen  gave equivalent definition of topological entropy which measures for how fast the solutions of dynamical system move part. For a linear map on , topological entropy is given by the sum of the logarithms of the operator’s eigenvalues with absolute value greater than 1, see  . Recently, Hoock generates for certain infinite-dimensional linear systems, see  . In particular, he also showed that topological entropy of a strongly continuous semigroup is given by sum of real parts of the unstable eigenvalues of the infinitesimal generator, if the unstable part is finite-dimensional. The main result of present paper is a generalization of several results for nonautonomous linear systems in the finite-dimensional case.
Now we introduce some basic notations for nonautonomous differential equations. Let the linear equation
where is the real matrix function which is uniformly bounded on . In this paper, we consider is a fundamental matrix solution of (1). For the basic theory of this Equation (1), we refer to the book of Dalecki et al.  . In order to describe topological entropy for (1), we introduce the concept of spanning and separated sets following  . For any , define a metrix on by
Let K be a compact subset of . For any , a subset is said to be an -spanning set of K, if for any there exists such that . Let denote the minimal cardinality of any -spanning set of K.
Analogously, a set is said to be an -separated set of K, if , , implies . Let denote the maximal cardinality of any -separated set of K.
Lemma 1.1. Let is a fundamental matrix solution above. Assume and be a compact set. We have that
Proof. Suppose S is the -separated set with maximal cardinality. By definition, if then for all . Therefore S is the -spanning, it means the first inequality hold. To prove the second one, we set R is the minimal -spanning set. Then we have
where is a ball, centre x and radius r. Let is the maximal -separated set. If for some then
It means (since the definition of -separated set) and hence the second inequality is proved.
By previous lemma, the following definition of topological entropy makes sense.
Definition 1.1. Let . For a compact set and is a fundamental matrix solution of (1), topological entropy for is given by
Remark 1.1. If is a constant matrix for all then the definition above coincide the definition of A.-M. Hoock (see  ), i.e. , The sum is taken over all eigenvalues of A with .
Remark 1.2. If is other fundamental matrix solution of (1) then . Indeed, by  there is a converse matrix C such that . If x is belong to -spanning set of a compact set K for then x is the same for . Hence, . Similarly, one also have . It is our purpose.
If we use A to present the Equation (1), by Remark 1.2, we define the topological entropy for (1), denote , as following
where X is some fundamental matrix solutions.
Remark 1.3. Since all norms on are equivalent so does not depend on the norm chosen.
We now give an outline of the contents of this paper. In Section 2, one gives the upper estimation for topological entropy for the class of bounded equations. In particular, we are going to show that one is less than nM where n is dimension of space and M upper bounded of for all . In Section 3, we concentrate the invariant property of topological entropy. As consequence, one shall prove that topological entropy of the periodic equations is equal to the sum of all positive Lyapunov characteristic exponents of them. Finally, Section 4, we shall show that topological entropy of (1) is equal to sum of positive Lyapunov characteristic exponents.
2. Estimation of Topological Entropy for Bounded Linear Equation
In this section we shall give the estimation of topological entropy for bounded linear equation. We shall begin with the following technique lemma.
Lemma 2.1. Let any . Then
Proof. It is clear that . Converse, we know that for any , one have and
Fix . Choose such that
Since there are infinity sets , so, we can choose such that
and is fixed set (i.e. for any k). Therefore . Similarly, we also can choose such that is fixed set. Conclusion,
Finally, any compact subsets K of can be covered by a finite number of balls of diameter and hence
which give the relation (2).
The following theorem is the main theorem in this section.
Theorem 2.1. Assume the Equation (1) has matrix function satisfies for all . Then
where n is a dimension of matrix .
Proof. Let m is the Lebesgue measure on , a fundamental matrix solutions of (1). First of all, we is proving the following claim
where we denote is the ball whose centre at a with radius r. Indeed, let K is a compact subset of with . If R is a -spanning set of K then
The last relation is true for all the compact sets K in . It means
To prove the converse inequality, suppose with is a arbitrary number such that . Suppose S is an -separated subset of K. Then
for all in K. The well-known result that
where is Euler’s gamma function, is the volume for ball of radius n. We have
Because the last inequality hold for all , by Lemma 2.1, we obtain
From (4) and (6), the desired our claim hold. For any , one have
It leads to
On the other hand, (see J. L. Dalecki  ), let , the last inequality becomes
Compare with the claim (3), the desired inequality hold.
3. Topological Entropy and the Transformations
Let the equation
where B is the real matrix function which is also uniformly bounded on . Let , are fundamental matrix solutions of (1) and (7), respectively. The solutions of the Equations (1) and (7) are said to be topological conjugate if there is a homeomorphism such that
for every and .
To start this section, we give the question: Is topological entropy invariant property with the topological conjugacy? The first, one considers the simple example. Let the two equations, namely A, B, corresponding,
where and . As in  , A.-M. Hoock shown that . On the other hand, by Theorem 2.50 in  two the equations above are topological conjugacy. Hence, topological entropy is not invariant property with the topological conjugacy. The following, we shall give a compare critical of topological entropy in term of homeomorphism h and a sufficient condition of homeomorphism h such that topological entropy is invariant.
Proposition 3.1. Let , are fundamental matrix solutions of (1) and (7), respectively. Assuming there exists homeomorphism satisfies . Then the following statements hold
1) h is a noncontraction (i.e. ) then ,
2) h is a nonexpanding (i.e. ) then .
Proof. Without lost the generation, we suppose that h is nonexpanding map. Let is a compact set in and S is a separated set of with Equation (1) which is has cardinality is equal to . Let , by definition of S, we obtain . Since the hypothesis of (1), one have estimate
Therefore, is -separated set of with Equation (7). Hence
In other word, . The proof of (2) is similar.
Corollary 3.1. If h is a isometric homeomorphism such that then
Remark 3.1. For the case of discrete, topological entropy is invariant to topological conjugacy, but it is no longer true for continuous case, even for the coefficient matrix is constant. This implies that topological entropy becomes more complex in continuous case. In other words, topological conjugacy cannot preserve the speed of the lose information for nonautonomous linear equations.
The following, we are going to consider property of topological entropy with topological equivalence.
We say that (1) and (7) are topological equivalence (see  ) if there exists a continuous function with the following properties
1) and as uniformly with respect to t,
2) , defined by is a homogeneous for each fixed t,
3) , defined by , is continuous and has property (1) also,
4) If is a solution of (1) then is a solution of (7).
Remark 3.2. Condition (4) implies the equality
Remark 3.3. A straightforward verification shows that topological equivalence is an equivalence relation in the class of nonautonomous equations.
The Equations (1) and (7) are said to be kinematically similar if there exists a continuous differential invertible matrix function (called a kinematic similarity) such that and are bounded and such that the transformation takes the solutions of (1) on to the solutions of (2).
Remark 3.4. If the Equations (1) and (7) are kinematically similar, then they are topological equivalence. Indeed, in the definition of topological equivalence it suffices to set where is the function realizing the kinematically similarity.
The following theorem presents the sufficient condition of topological equivalence which prevents topological entropy.
Proposition 3.2. Let (1) and (7) are topological equivalence with the homeomorphism satisfy
where are scalar bounded function on and positive constants. Then .
Proof. Let , are fundamental matrix solutions of (1) and (7), respectively. Suppose be a compact set and is a minimal -spanning of . Then is a minimal -spanning of . Indeed, let any , by definition of spanning set, there exits such that . We have following estimation
where . It implies
By the similar proof above, we also have . The proposition is complete.
Remark 3.5. It is clear that if (1) and (7) are kinematically similar then they satisfy all hypothesis of previous proposition with , , (where is kinematic similarity). Therefore the class of kinematically similar nonautonomous equations is invariant topological entropy.
Corollary 3.2. If (1) is periodic equation then
where the sum take all the positive Lyapunov characteristic exponents of that equation.
Proof. By Theorem 2.3.1 in  and from previous remark, we obtain where are a constant matrix. On the other hand, by  , where the sum takes all the positive eigenvalues of B. Using  again, we have the complete proof.
4. Topological Entropy and Lyapunov Exponents
In this section, we show that topological entropy of the Equation (1) is equal to sum of positive Lyapunov characteristic exponents.
Given a fundamental matrix solution X of (1), consider the quantities
where denotes the ith standard unit vector. When is minimized with respect to all possible fundamental matrix solutions, then the are called the Lyapunov exponents, or Lyapunov characteristic numbers, and the corresponding fundamental matrix solution is called a normal basic.
In this section, we can always work with a normal basis which has ordered Lyapunov exponents
With these definitions we get the following theorem.
Proof. Let fixed. Assume that we can choose a fix point such that K is covered by a box
where is the ith unit vectors. Suppose the fundamental matrix solution is arranged in the order a increase of the Lyapunov exponents. For each , we consider the finite subset of which is given by
Claim 1. The subset is an -spanning set of K.
Proof of Claim 1.
For any then x can be written the form
for some . For every small enough, one choose such that
We now set
then for any we get
From the last equation and definition of Lyapunov characteristic exponents, one obtain
Choose small enough such that for all . The last inequality implies
Hence, the Claim 1 is proved.
It is clear that -spanning set have
Therefore, we have following estimation
Since is a arbitrary small positive constant, we have
To order the reverse inequality, let is a ball with centre x and radius . Denote
We would prove the following claim.
Claim 2. The subset is an -separated set of the .
Proof of Claim 2.
Let two distinct points in , namely
for some . Let we get
Hence, we obtain
Since is supermum of , take all so
Combining (8) and (9), we conclude the proof.
The first author was supported in part by the VNU Project of Vietnam National University No. QG101-15.
How is the topological entropy for the class of unbounded linear equations on ?