Abstract
Expansiveness is very closely related to the stability theory of the dynamical systems. It is natural to consider various types of expansiveness such as countably-expansive, measure expansive, N-expansive, and so on. In this article, we introduce the new concept of countably expansiveness for continuous dynamical systems on a compact connected smooth manifold M by using the dense set D of M, which is different from the weak expansive flows. We establish some examples having the countably expansive property, and we prove that if a vector field X of M is stably countably expansive then it is quasi-Anosov.
MSC:
37C20; 37C05; 37C29; 37D05
1. Introduction
Let X be a compact metric space with a metric d and be a homeomorphism. Utz [] introduced a dynamic property, which is called expansiveness. It means that, if two orbits stay within a small distance, then the orbits are the same. That is, a homeomorphism f is expansive if there is an expansive constant such that for any there is satisfying From the definition of the expansiveness, it is possible to consider the set
We can easily check that f is expansive if and only if for all
Now, we have a natural question:
Definition 1
([] Definition 2.8). Given a homeomorphism f of X is N-expansive on if there is an expansive constant such that has at most N elements for all If . Then, we say that f is N-expansive.
It is easy to see that if f is expansive then f is N-expansive. Now, we introduce another notion of expansiveness, which is a general notion of expansiveness.
Definition 2
([] Definition 1.6). We say that a homeomorphism f of X is countably expansive if there is an expansive constant such that for all the set is countable.
Note that the relationship with among those notions is
expansive ⇒ N-expansive ⇒ countably expansive.
On the other hand, from the stochastic point of view, Morales and Sirvent [] introduced a general notion of expansiveness by using a measure. For the Borel -algebra on X, we denote the set of Borel probability measures on X endowed with the weak topology. Let be the set of nonatomic measure
Definition 3
([] Definition 1.3). We say that a homeomorphism f of X is μ-expansive if there exists an expansive constant such that for all We say that f is measure expansive if it is μ-expansive
In among the notions, a remarkable notion is measure expansiveness (which was introduced by Morales []). It is exactly same as countably expansiveness (see []). That is, Artigue and Carrasco-Olivera [] considered a relationship between the measure expansiveness and the countably expansiveness.
Remark 1
([] Theorem 2.1). Let be a homeomorphism. Then,
f is countably expansive ⟺ f is measure expansive.
Let M be a compact connected smooth manifold, and let Diff be the space of diffeomorphisms of M endowed with the topology. Denote by d the distance on M induced from a Riemannian metric on the tangent bundle In dynamical systems, the concept of expansiveness [] is a useful notion for studying stability theory. In fact, Mañé [] showed that if a diffeomorphism f of M is stably expansive then it is quasi-Anosov. Here, we say that f is quasi-Anosov if, for all the set is unbounded.
Later, many mathematicians studied stability theory using the various types of expansiveness [,,,]. For instance, Moriaysu, Sakai, and Yamamoto [] showed that if a diffeomorphism f of M is stably measure expansive then it is quasi-Anosov. The result is a generalization of the result of Mañé [].
On the other hand, it is very important to extend from diffeomorphisms to vector fields (flows). In fact, many researchers studied the various aspects of flows, such as thermodynamics-Hamiltonian systems [], nonlinear systems [], and chaos systems [,].
From the result of [], Moriyasu, Sakai, and Sun [] extended expansive diffeomorphisms to vector fields about the stably point of view. That is, they showed that if a vector field X is stably expansive then it is quasi-Anosov. Lee and Oh [] showed that if a vector field X is stably measure expansive then it is quasi-Anosov (We refer to the basic definitions related to the vector fields below.).
In this article, we introduce another type of countably expansive vector fields which is different than weak expansiveness in []. In addition, we establish some examples of the countably expansiveness for homeomorphisms and flows, such as shift map and suspension flow by applying the rotation map on the circle. Moreover, we prove that if a vector field X of a compact connected manifold M is stably countably expansive, then it is quasi-Anosov which is a general result of Moriyasu, Sakai, and Sun []. Furthermore, we have that if a vector field X of a compact connected manifold M is stably expansive, weak expansive, and countably expansive then it is quasi-Anosov.
2. Countably Expansiveness for Suspension Flows
In this paper, we focus on countably expansiveness which is defined as the following remark.
Remark 2.
In general, according to the Baire Category Theorem, there is a dense subset in a compact metric space Especially, we consider a dense subset to define the countably expansiveness on this space
Definition 4.
We say that a homeomphism is countably expansive if there is such that for all the set
is countable, where is the closure of A.
Example 1.
Assume that X is a separable space and is a homeomorphism. Then, it is clear that f is “countable expansive” (according to Definition 4).
Example 2.
Let be an irrational rotation map. For all , we consider the set
where is the rational numbers. Then, it is clear that and : countable, denoted by the cardinality of a set Thus, f is countably expansive. However, and is not a finite set. Therefore, the map is neither expansive nor N-expansive.
Symbolic systems can be used to “code” some smooth systems. Indeed, to study of symbolic dynamics is the research of a specific class of the shift transformation in a sequence space. In addition, it provides more motivation of the relationships between topological and smooth dynamics. The properties of symbolic dynamical systems give a rich source of examples and counterexamples for topological dynamics and ergodic theory.
The set of all infinite sequences of 0s and 1s is called the sequence space of 0 and 1 or the symbol space of 0 and 1 is denoted by More precisely, We often refer to elements of as points in Shift map is defined by
In short, the shift map “deletets” is the first coordinate of the sequence, for example (for more details, see []).
Definition 5
([] Definition 11.2). Let and be points in We denote the distance between s and t as and define it by
Since is either 0 or 1, we know that
The shift map is continuous; it is clear that two points are close if and only if their initial coordinates are same. The more the coordinates are the same before they are different, the closer they are to each other. Then, we know that the set of periodic points of the shift map is dense in (the shift map has periodic points of period n). We put the set
then where is a dense set of In fact, we can check the above facts by the following example.
Example 3.
Let and Then,
is countable.
Proof.
Let and Then, we have two cases as follows.
- (i)
- for implies
- (ii)
- If , then for implies there exists such that That is, the jth coordinate of and have at least one different coordinate components.
For two cases, we consider the cardinality of where j is countably many. This means that is a countable set. □
On the other hand, we need to check some properties of countably expansive homeomorphism, which are used to prove the lemmas and theorems below.
Lemma 1.
If a homeomorphism f of X is countably expansive, then is countably expansive for a closed subset
Proof.
Since f is countably expansive, there exists such that is countable set for all Let A be a closed subset of Then, is dense in X. If satisfying for all and for all then This means that As is countable, is countably expansive. □
Example 4.
Let f be the identity of the interval and It is clear that is expansive but f is not expansive in the whole interval. Thus, the converse of Lemma 1 cannot hold.
Lemma 2.
Let be a homeomorphism. Then,
- (a)
- f is countably expansive if and only is countably expansive, for some
- (b)
- If f is the identity map , then f is not countably expansive.
Proof of (a).
(⇒) For fixed since X is compact we can choose such that if then We have Since is countable, is countable as well. Therefore, is countably expansive.
(⇐) Let be dense in X and be the countably expansive constant for for some For all Let Then,
This means that f is countably expansive. □
Proof of (b).
Suppose that f is the identity map Then, there exists such that for any As we know that the set is uncountable, f is not countably expansive. □
Remark 3.
In Lemma 2, (b) says that if the identity is countable expansive then the space is countable (there are countable compact metric spaces).
Lee, Morales, and Thach [] characterized the countably expansive flows in measure-theoretical terms, which is extended the result of [] in the discrete case, called weak expansive flows. They showed that a flow is countably expansive if and only if the flow is weak measure expansive.
Definition 6
([] Definition 1.1). A flow ϕ on a compact metric space X is countably expansive if there is an expansive constant such that for any and there is an at most countable subset satisfying where
Here, denotes the set of continuous maps with and
Now, we introduce the new notion of countably expansive flows by using a dense subset D of X and consider the examples showing the countably expansive property, very well. A continuous flow of X such that
- satisfying
- for and
Denote by
Definition 7.
We say that a flow ϕ of X is countably expansive if there exist an expansive constant and dense subset D of X such that is a countable set, where
and denotes the set of increasing continuous maps with
Now, let be a continuous function and consider the space
with for each The suspension flow over with height function is the flow on defined by
More precisely, for all and for all For all and there is a unique such that we set
The Bowen–Walters distance (Definition 2 of []) makes a compact metric space where a neighborhood of a point contains all the points of where is small and w is close to With respect to the topology generated, is a homeomorphism on for all
Carrasco-Olivera and Morales [] extended the concept of expansive measure from homeomorphism [] to flows. They (respectively []) showed that a homeomorphism of a compact metric space is measure expansive (respectively, countable expansive) if its suspension flow is. The following theorem says that the case of countably expansiveness, which is defined in this paper, is also satisfied.
Theorem 1.
Let be a homeomorphism. Then, f is countably expansive if and only if there is a continuous map such that the suspension flow on over f with height function τ is countably expansive.
Proof.
It is enough to show that “only if” part. Let be given by for all Then, the quotient space associated with X and the suspension flow on given by
Now, we claim that is countably expansive. It is sufficient to show that there are a constant and a dense subset of such that
for all and some Since and D is a dense subset of Thus, is a countable set. Therefore, the suspension flow on over f with height function is countably expansive. □
By the following examples, it is easy to see that a suspension flow over an irrational rotation map on the unit circle is countably expansive by applying the Theorem 1.
Example 5.
Consider a flow ϕ on the unit circle given by
Then ϕ is countably expansive.
In addition, we can see that the following example satisfies Theorem 1.
Example 6.
If is an irrational rotation map then there is a continuous map such that the suspension flow on over f with height function τ is countably expansive.
On the other hand, we can find a dense subset of as following remark.
Remark 4.
As we know that
is a dense subset of we can check that is a dense subset of Fix let in Then, there exists such that for all Since
. Therefore
In fact there exists such that for all then This means that is a dense subset of
Moreover, we can show that Theorem 1 holds by using the above remark for the case of the suspension flow over a shift map as following example.
Example 7.
If is countably expansive if and only if there is a continuous map such that the suspension flow on over σ with height function τ is countably expansive.
Proof.
Let be given by for all Then, the quotient space corresponding to and the suspension flow on given by
Now, we claim that is countably expansive. It is enough to show that there are a constant and a dense subset of such that
for all and some Since and D is a dense subset of
Thus is a countable set. Therefore, the suspension flow on over with height function is countably expansive. □
3. Stably Countably Expansive Vector Fields
Recall that M is a compact connected smooth manifold, d is the distance on M induced from a Riemannian metric on the tangent bundle Denote by the set of all vector fields of M endowed with the topology. Then, every generates a flow satisfying for all , and for any . Here, is called the integrated flow of Throughout this paper, for , denote the integrated flows by , respectively.
Note:To study of dynamical systems, the properties of orbits (or points) are important: singular, periodic, non-wandering, etc. If a flow has a periodic orbit or singularity, then it causes a chaos phenomenon (for example, Geometric Lorenz attractor). This means that we cannot control the system. Especially, the countably expansive flow which we present in this paper does not have a singularity. Thus, we could investigate the stability of countably expansive flows.
For , denote by the orbit of the flow (or X) through x. A point is singular of X if denotes the set of singular points of It is said that a point p is periodic if for some but for all denotes the set of periodic points of A point p is regular if and The set of non-wandering points of X, denoted by then we can see that
A flow of M is expansive if for given there is a expansive constant such that if satisfying for some and all then , where denotes the set of increasing continuous maps fixing
Definition 8.
We say that is countably expansive if there exists such that
where
First, we can check some properties of countably expansive flows as following lemmas.
Lemma 3.
If is countably expansive, then Sing is totally disconnected.
Proof.
Assume that Sing is not totally disconnected. Take For any let be a closed small arc with two end points x and y such that the length of is less than Let be an expansive constant. We can take a local chart satisfying Then, Thus, we consider the dense set and we can easily see that and D is uncountable. For any
This means that X is not countably expansive. This contradicts to complete the proof. □
We can see that the singular points of countably expansive flows are isolated by the below lemma.
Lemma 4.
Let If the flow is countably expansive, then every singular points of X is isolated.
Proof.
Suppose that there exist Sing which are not isolated. Let be a closed small arc with two endpoints x and By Lemma 3, this is a contradiction. Thus, every singular points is isolated. □
A closed -invariant subset is hyperbolic if there exist constants , and a splitting satisfying the tangent flow has invariant continuous splitting and
for and We say that is Anosov when M is hyperbolic for
We say that a vector field X is if is dense in and is hyperbolic. For Axiom A vector field we know that is equal to the union of each basic set of Note that the basic set is closed, invariant, and transitive. A collection of basic sets of X is called a cycle if, for each there exists such that and We say that a vector field X has no cycle if there exist no cycles among the basic sets of
For any hyperbolic periodic point x of X, the sets
are the stable manifold and unstable manifold of x, respectively. For Axiom A vector field we say that X has the quasi-transversality condition if for any .
The exponential map defined by for all where Let For any we set
where for
Let be the normal bundle on Then, we present a linear Poincaré flow for X on by
where is the natural projection along the direction of and is the derivative map of We say that is quasi-Anosov if for then .
Definition 9.
We say that the integrated flow of is stably countably expansive if there is a neighborhood of X such that the integrated flow of is countably expansive.
The main theorem of this paper stated as follows.
Theorem 2.
If a vector field X is stably countably expansive, then it satisfies Axiom A without cycle condition.
Now, let us prove the above theorem. To show this, we first need following lemma.
Lemma 5.
Suppose that Sing and For the Poincaré map let be a neighborhood of X and given Then, there are and such that for a map with there is satisfying
where and is the Poincaré map of
Proof.
By Lemma of []. □
Denote as the set of with the property that there is a neighborhood of X such that every PO is hyperbolic for It was proved by [] that if and only if X satisfies Axiom A without cycle condition.
Proof of Theorem 2.
Let X be stably countably expansive. Then, the proof is completed by showing Suppose there exists Then, there are and Y has a non-hyperbolic periodic point
Let and be the Poincaré map of at As p is a non-hyperbolic fixed point of there is an eigenvalue of with Let and be given by Lemma 5 for and Then, for the linear isomorphism there is satisfying
Here, g is the Poincaré map of Since the eigenvalue of is we can take a vector associated to such that and Then,
Put and . Then, is an invariant small arc such that
and so where is the time T-map of the flow Since is the identity on , is not countably expansive. Thus, this contradict to that X is stably countably expansive. Therefore, we completed the proof. □
For given take a constant a map such that and for The Poincaré map is given by
For given we denote by the set of diffeomorphisms satisfying supp and Here, is the metric, is the identity map and supp where it differs from
Lemma 6.
Suppose that and Sing For the Poincaré map and let be neighborhood of X and Then, there is with the property that for any there exists such that
where is the Poincaré map of
Proof.
By Remark 2 of []. □
Theorem 3.
If a vector field X is stably countably expansive, then X is quasi-Anosov.
Proof.
It is enough to show that if the flow of is stably countably expansive then X satisfies the quasi-transversality condition by applying Theorem A of [] and Theorem 2.
Assume that there exists X such that it does not satisfy the quasi-transversality condition. Then, there exists such that
and thus we have By Lemma 6 with a small perturbation of X at we can construct and an arc centered at There exists a local chart such that and is diffeomorphic to Now, we consider the set where is the Poincaré map defined by That is,
We can check that the set is uncountable, easily. Therefore, is not countably expansive. The contradiction completes the proof. □
4. Conclusions
The theory of dynamical systems is motivated by the search of knowledge about the orbits of a given dynamical systems. To describe the dynamics on the underlying space, it is usual to use the notion of expansiveness. In the various type of expansiveness for a homeomorphism of a compact metric space Artigue and Carrasco-Olivera proved that a homeomorphism is countably expansive if and only if f is measure expansive (Theorem 2.1 []).
In this article, we extend the countably expansiveness to the continuous dynamical systems. However, there is a problem to define the countably expansiveness for flows caused by reparameterization for each point in of Definition 8. Therefore, we define a suitable concept of countably expansiveness for flows as an improvement of the problem. By using this concept, we apply the results for expansiveness and measure expansiveness to the countably expansive flows. More precisely, we prove that, if a vector field X is stably countably expansive, then it satisfies Axiom A without cycle condition. Furthermore, it is quasi-Anosov.
Author Contributions
Writing–original draft, J.O.; Writing–review and editing, M.L.
Funding
The first author is supported by NRF-2017R1A2B4001892. The second author is supported by the National Research Foundation of Korea (NRF) grant funded by the MEST 2015R1A3A2031159 and NRF 2019R1A2C1002150.
Acknowledgments
The authors wish to express their appreciation to Keonhee Lee for his valuable comments.
Conflicts of Interest
The authors declare no conflict of interest.
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