On the Topological Entropy of Nonautonomous Differential Equations ()
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. [1] as an invariant of topological conjugacy. Later, Bowen [2] 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 [2] . Recently, Hoock generates for certain infinite-dimensional linear systems, see [3] . 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
(1)
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. [4] . In order to describe topological entropy for (1), we introduce the concept of spanning and separated sets following [2] . 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 [3] ), 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 [5] 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.
Assuming
.
Lemma 2.1. Let any
. Then
(2)
Proof. It is clear that
. Converse, we know that for any
, one have
and
Fix
. Choose
such that
Then
and so
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
(3)
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
It implies
Hence,
The last relation is true for all the compact sets K in
. It means
(4)
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
(5)
where
is Euler’s gamma function, is the volume for ball of radius n. We have
Therefore,
Because the last inequality hold for all
, by Lemma 2.1, we obtain
(6)
From (4) and (6), the desired our claim hold. For any
, one have
Therefore,
Hence,
It leads to
(by (5))
or
On the other hand,
(see J. L. Dalecki [4] ), let
, the last inequality becomes
Compare with the claim (3), the desired inequality hold.
3. Topological Entropy and the Transformations
Let the equation
(7)
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 [3] , A.-M. Hoock shown that
. On the other hand, by Theorem 2.50 in [5] 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
It implies
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 [6] ) 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
Hence,
or
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 [5] and from previous remark, we obtain
where
are a constant matrix. On the other hand, by [3] ,
where the sum takes all the positive eigenvalues of B. Using [5] 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.
Theorem 4.1.
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
(8)
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
(9)
Combining (8) and (9), we conclude the proof.
Acknowledgements
The first author was supported in part by the VNU Project of Vietnam National University No. QG101-15.
Open Questions
How is the topological entropy for the class of unbounded linear equations on
?