1. Introduction
Inverse shadowing was introduced by Corless and Pilyugin [1] and also as a part of the concept of bishadowing by Diamond et al. [2]. Kloeden, Ombach and Pokroskii [3] defined this property using the concept of
-method. One can also see [4-7] for more information about the concept of
-method. Authors in [8] studied on locally genericity of weak inverse shadowing with respect to class
. For flows, there are lots of existing work on finding the minimal sets in a systems with shadowing property. See for example [9-12]. In this paper we study diffeomorphisms with weak inverse shadowing property with respect to class as
and
. First we show that an
-stable diffeomorphism
has weak inverse shadowing property with respect to classes of continuous method
and
(Theorem 1) and some
-stable diffeomorphisms have weak inverse shadowing property with respect to classes
(Theorem 2). In addition we study relation between minimality and weak inverse shadowing property with respect to class
and show that a chain transitive homeomorphism
on compact metric space
is minimal if and only if it has weak inverse shadowing property with respect to class
(Theorem 3). Finally we study relation between positively expansive and inverse shadowing property with respect to class
and show that if
has inverse shadowing property with respect to class
, then
is not positive expansive (Theorem 4).
Let
be a compact metric space and let
be a homeomorphism (a discrete dynamical system on
). A sequence
is called an orbit of
, denote by
, if for each
,
and is called a
-pseudo-orbit of
if ![](https://www.scirp.org/html/12-7400762\45b5488a-b8f3-463b-9748-7e9a74efd5ac.jpg)
Denote the set of all homeomorphisms of
by
. In
consider the complete metric
![](https://www.scirp.org/html/12-7400762\ac79480e-beb6-42b4-bd75-f0dd54cef5e2.jpg)
which generates the
-topology.
Let
be the space of all two sided sequence
with elements
, endowed with the product topology. For
let
denote the set of all
-pseudo orbits of
.
A mapping
is said to be a
-method for
if
, where
is the 0-component of
. If
is a
-method which is continuous then it is called a continuous
-method. The set of all
-methods (resp. continuous
-methods) for
will be denoted by
(resp.
). If
is a homeomorphism with
, then
induces a continuous
-method
for
defined by
![](https://www.scirp.org/html/12-7400762\36cf36fb-5edd-48b3-8514-5d5346cbbf54.jpg)
Let
denote the set of all continuous
- methods
for
which are induced by
with
.
Let
and
, a homeomorphism
is said to have the inverse shadowing property with respect to the class
,
, c, h, in
if for any
there is
such that for any
-method
in
and any point
there exists a point
for which
![](https://www.scirp.org/html/12-7400762\ef72c16c-72d4-4c0c-89fe-9f51f7baff55.jpg)
A homeomorphiosm
is said to have weak inverse shadowing property with respect to the class
,
, c, h, in
if for any
there is
such that for any
-method
in
and any point
there exists a point
for which
![](https://www.scirp.org/html/12-7400762\03721cc6-a6ec-4d50-9dbe-9f1167939cf0.jpg)
Fix
. A continuous
-method of class
for the diffeomorphism
is a sequence
, where any
is a continuous mapping
such that
![](https://www.scirp.org/html/12-7400762\fbf4febb-26d8-4d84-862e-b5db714a2025.jpg)
A sequence
is a pseudo-orbit generated by a continuous
-method
of a class
if ![](https://www.scirp.org/html/12-7400762\4aa72182-2241-450c-a7b1-3961055fa2ff.jpg)
Fix
. A continuous
-method of class
for the diffeomophism
is a sequence
, with
for
and such that any
is a continuous mapping
with the property
![](https://www.scirp.org/html/12-7400762\815ca44f-3c77-4019-b940-63c9cd6aea75.jpg)
A sequence
is a pseudo-orbit generated by a continuous
-method
of class
if
![](https://www.scirp.org/html/12-7400762\2d4fbe4a-0374-4541-bf37-24959a6d80c4.jpg)
If a sequence is generated by
or
we briefly write
.
A diffeomorphism
is said to have (weak) inverse shadowing property if for any
and
there exists
such that, for any continuous
-method
, we can find a pseudo-orbit
satisfying the inequalities
![](https://www.scirp.org/html/12-7400762\92d2b6f3-3293-4b85-b007-dc1091d20123.jpg)
![](https://www.scirp.org/html/12-7400762\956a602a-39d9-473d-b6a1-9bd555e84f8e.jpg)
Pilyugin [5] showed that a structurally stable diffeomoriphism has the inverse shadowing property with respect to classes of continuous method,
and
. He also showed that any diffeomorphism belonging to the
-interior of the set of diffeomorphisms having the inverse shadowing property with respect to classes of continuous method,
and
is structurally stable.
2. Diffeomorphisms with Weak Inverse Shadowing Property with Respect to Class θs, θc and ![](https://www.scirp.org/html/12-7400762\a7dce034-1cf7-49be-834c-b502692677b0.jpg)
In this section we show that an
-stable diffeomorphism
has the weak inverse shadowing property with respect to classes of continuous methods
and
and if we impose some condition on an
-stable diffeomorphism, then it has weak inverse shadowing property with respect to classes
.
Theorem 1 If a diffeomorphism
is
-stable, then it has the weak inverse shadowing property with respect to both classes
and
.
Before proving this main result, let us briefly recall some definitions. A diffeomorphism
is called
-stable if there is a
-neighborhood
of
such that for any
,
is topologically conjugate to
.
A diffeomorfphism
is called an Axiom
system if
is hyperbolic and if
.
Axiom
and no-cycle systems are
-stable [13].
Let
be an Axiom
diffeomorphism of
. By the Smale spectral Decomposition Theorem, the nonwandering set
an e represented as a finite union of basic sets
.
![](https://www.scirp.org/html/12-7400762\3a7b3513-e782-4ced-acd0-7da9ae6533dd.jpg)
In the proof of theorem 1 in [5], Pilyugin has used the following statement.
If a
-diffeomorphism
satisfies Axiom
and the strong transversality condition, then there exist constants
and
and linear subspace
,
of
for
such that
![](https://www.scirp.org/html/12-7400762\7dfa920b-04b2-46bd-9a80-73824af494de.jpg)
and
(1)
(2)
if
and
are the projectors in
onto
parallel to
and onto
parallel to
, respectively, then
(3)
(here
is the operator norm).
Conditions (1), (2) and (3) play a basic role in the proof of theorem 1 in [5]. If
is a basic set then we can see for every
, conditions (1), (2) hold. Since
is bounded for
, standard reasening shows (see, for example, Lemma 12.1 in [14]) that there exists a constant
for which inequalities (3) hold. Hence similar to the proof of theorem 1 in [5],
has the inverse shadowing property with respect to classes
and
on
. The following two propositions are well known (proposition 1 is the classical Birkhoff theorem [13], for proofs of statements similar to proposition 2, see [15], for example).
proposition 1 Let
be a homeomorphism of a compact topological space
and
be a neighborhood of its nonwandering set. Then there exists a positive integer
such that
![](https://www.scirp.org/html/12-7400762\b6b011b6-8166-493e-9dab-3b3d1b892eda.jpg)
for every
, where
is the cardinality of a set
.
In the following proposition, we assume that
is an
-stable diffeomorphism of a closed smooth manifold.
proposition 2 If
is a basic set, then for any neighborhood
of
there exists neighborhood
with the following property: if for some
and
,
, then
for
.
Lemma 1 Let
be an
-stable diffeomorphism and
be the Smale Spectral Decomposition. Let
be a neighborhood of
for
. Then for any
there exists
and
for some
, such that
![](https://www.scirp.org/html/12-7400762\e01bede9-180b-4f4b-a337-225fd6676bf6.jpg)
and similarly there exists ![](https://www.scirp.org/html/12-7400762\4aa35810-1209-41e6-b533-48ec69ddb45e.jpg)
![](https://www.scirp.org/html/12-7400762\81facf35-8362-460c-a724-2870e097b774.jpg)
Proof. Suppose that the lemma is not true for some
. Let
be a neighborhood of
as in proposition 2. Proposition 1 shows that there exists
such that
for some
. By assumption there exists
such that
. By proposition 2,
for
. Thus using proposition 1, there exists
such that
for some
and there exists
such that
. By proposition 2,
for
. This process show that
![](https://www.scirp.org/html/12-7400762\1aead5cd-80f0-4366-acb3-3bb85114f292.jpg)
contradicting proposition 1. Proof of
![](https://www.scirp.org/html/12-7400762\52798307-7de2-403d-999a-0113498cf288.jpg)
is similar.
Proof of theorem 1. Let
and
be arbitrary. Let
be a neighborhood of
for
, such that shadowing property hold for them. By lemma 1 there exists a positive number
, such that
for some
. Since
is compact, there exist
, such that
where
is as in the shadowing theorem for hyperbolic set. So ![](https://www.scirp.org/html/12-7400762\310c90fc-bde2-44e3-860d-903a4fbd993e.jpg)
is a periodic
-pseudo-orbit of
in
. By shadowing theorem for hyperbolic sets, there is
which
-shadows
. This shows that
![](https://www.scirp.org/html/12-7400762\73e70627-59ea-43b7-a79c-322b6be18df2.jpg)
But
has the inverse shadowing property with respect to classes
and
. Thus there exists
such that for any continuous
-method
, we can find a pseudo-orbit
satisfying
![](https://www.scirp.org/html/12-7400762\fd356666-b553-47d0-a438-ff287f79d264.jpg)
Inequalities
and
show that
. This complete the proof of theorem 1.
Theorem 2 Let
be an
-stable diffeomorphism and
be the Smale Spectral Decomposition such that
be fix point sources or sinks. Then
has the weak inverse shadowing property with respect to class
in
, where
is set of fix points of
.
Proof. Let
and
be arbitrary that is not fix point and
be open neighborhoods of
respectively with diameter less than
. Lemma 1 shows that there exists
and
and
for some
, such that
![](https://www.scirp.org/html/12-7400762\7afb907b-9604-4819-b491-49f915074a20.jpg)
and
![](https://www.scirp.org/html/12-7400762\5b17e21c-ae09-499d-8811-1cfb225c387e.jpg)
Note that
is a neighborhood of fix point sink and
is a neighborhood of fix point source. Choose
![](https://www.scirp.org/html/12-7400762\c4939c32-b7f7-4c4d-95d1-90bb3842c320.jpg)
such that
![](https://www.scirp.org/html/12-7400762\fbb836f9-9f54-4182-8d10-2d5c01033ea2.jpg)
for every
with
, where
![](https://www.scirp.org/html/12-7400762\b8ca295a-ca2b-40c0-aaa5-6d80ac653f7b.jpg)
This shows that if
is a
-pseudo orbit and ![](https://www.scirp.org/html/12-7400762\c2c9fc47-447f-4ab7-b8b2-143222bdde6e.jpg)
then
. there exists
such that
(1)
and
(2)
Choose
such that if
then
![](https://www.scirp.org/html/12-7400762\08416bd3-c80d-4496-b17d-401ec37c8688.jpg)
And also
for
.
So for any
-pseudo orbit
![](https://www.scirp.org/html/12-7400762\9b0da2bc-55a0-41b1-923d-94259ddbe1fb.jpg)
we have
(3)
Now for any (
)-method
, by regarding the process of choosing
and (4), (5), (6) we have
, and this completes the proof of theorem 2.
The following example shows that an
-stable maybe has not the weak inverse shadowing property with respect to class
in its fix point.
Example. Represent
as the sqare
, with identified opposite sides. Let
be a diffeomorphism with the following properties:
The nonwandering set
of
is the union of 4 hyperbolic fixed points, that is,
, where
is a source,
is a sink, and
are saddles;
![](https://www.scirp.org/html/12-7400762\92546652-459e-42d4-92cc-2a92d1162059.jpg)
where
and
are the stable and unstable manifolds, respectively.
There exist neighborhoods
of
such that
for ![](https://www.scirp.org/html/12-7400762\0e4ab8e9-82a7-4962-9010-5a077abb2080.jpg)
The eigenvalues of
are
with
, and the eigenvalues of
are
with
.
Plamenevskaya [16] showed that
has the weak shadowing property if and only if the number
is irrational. Note that
does not have the shadowing property. We can see that
does not have the weak inverse shadowing property with respect to class
as well (Note that the number
is not necessary irrational). For any
, let
be the number of the weak inverse shadowing property of
. Construct a
-method as following:
![](https://www.scirp.org/html/12-7400762\b61ad690-5cbd-4d9b-8258-8b7013c3e6fb.jpg)
where
and
.
For every
, define
![](https://www.scirp.org/html/12-7400762\9d03c755-504f-43f6-80b8-1bd708074c1b.jpg)
3. Relation between Minimality and Weak Inverse Shadowing Property with Respect to Class ![](https://www.scirp.org/html/12-7400762\b68a27ae-762c-48cd-a311-e8a45edc08e8.jpg)
A homeomorphism
is called minimal if
, A closed, implies either
or
. It is easy to see that
is minimal if and only if
for every
.
A homeomorphiosm
is said to be chain transitive if for every
and
there are
-pseudoorbits from
to
and from
to
.
The following example shows that there exists homeomorphiosms
with inverse shadowing property with respect to class
which is not minimal.
Example. Let
with metric
![](https://www.scirp.org/html/12-7400762\d5d360a7-79f3-42e0-a136-b2131b596a9e.jpg)
Let
be a permutation of the set
for some
. Let
if
, and
otherwise.
is a homeomorphism and every point of
is a periodic point for
. We claim
has weak inverse shadowing property with respect to class
.
Proof of claim. Given
choose
such that
.
if and only if
where
. Let
and
be a
-method.
Let
, then
implies
for
and hence by definition of
,
for
. Also
implies
for
.
Hence if
and
then
for
, and so
.
Using this procedure we will get ![](https://www.scirp.org/html/12-7400762\50c0dd81-7240-4af6-9dc2-e0dbcf9a6ac6.jpg)
for
. A similar reasoning with having in mind that
is a homeomorphism proves that ![](https://www.scirp.org/html/12-7400762\1908e007-f405-4e51-960c-50fa6de201f8.jpg)
for
. Hence
for
and
has inverse shadowing property with respect to
. It is easy to see that
is not minimal.
Theorem 3 Let
be a chain transitive homeomorphism on compact metric space
. Then
is minimal if and only if
has weak inverse shadowing property with respect to class
.
Proof. Suppose that
has weak inverse shadowing property with respect to class
and
. Let
be an open set in
. Choose
and
such that
. There is
such that for each
-method
, there is
such that
![](https://www.scirp.org/html/12-7400762\2bb75115-de30-47e9-8545-04649f4d61cb.jpg)
For every
, there is a
-chain,
from
to
. Consider
![](https://www.scirp.org/html/12-7400762\40ed1fb5-f195-4470-bd73-f8a15a48bf14.jpg)
as a
-pseudo-orbit, such that it’s 0-component be
. Construct a
-method
such that
.
Hence there is
such that
, and so
for some
. Therefore
. This shows that each orbit of
is dense in
and so
is minimal. The converse i.e. to see that each minimal homeomorphism has weak inverse shadowing property with respect to class
, is obvious.
4. Relation between Expansivity and Inverse Shadowing Property with Respect to Class ![](https://www.scirp.org/html/12-7400762\4898ffc2-c232-4855-afd3-53acd36e6b13.jpg)
A homeomorphism
on metric space
is said expansive if there exists constant
such that for every
there exists integer number ![](https://www.scirp.org/html/12-7400762\e95bfa45-d9a4-4378-a817-7e282df6e1c1.jpg)
such that
.
Theorem 4 If homeomorphism
on metric space
has the inverse shadowing property with respect to class
, then
is not expansive.
Proof. Suppose that
is expansive and has the inverse shadowing property with respect to class
. Let
be as in definition of expansivity and
be such that for any
-method
in
and any point
there exists a point
for which
![](https://www.scirp.org/html/12-7400762\4e280614-5ac9-43b7-82e5-903989a8f10d.jpg)
Let
be arbitrary. Choose
such that
and
. Construct a
-method
as following.
For any
define
![](https://www.scirp.org/html/12-7400762\63bde9cb-a920-471c-880a-31f0e1456903.jpg)
and
![](https://www.scirp.org/html/12-7400762\a36b4b18-453a-404c-9932-9cc8f53a82f5.jpg)
Since
has the inverse shadowing property with respect to class
, for
there exists
such that
![](https://www.scirp.org/html/12-7400762\22591157-e096-4fd6-a9c2-5f1b69d76a7d.jpg)
By regarding to choose of
-method
, we have
![](https://www.scirp.org/html/12-7400762\7f9142e2-3d81-45e5-beed-99e9625359e6.jpg)
for some
, that contradicts the expansivity of
. This completes the proof of theorem.
5. Conclusion
In this paper we showed that an
-stable diffeomorphism
has the weak inverse shadowing property with respect to classes of continuous method
and
and some of the
-stable diffeomorphisms have weak inverse shadowing property with respect to classes
. In addition we studied relation between minimality and weak inverse shadowing property with respect to class
and relation between expansivity and inverse shadowing property with respect to class
.