Uniform Convergence and Dynamical Behavior of a Discrete Dynamical System ()
1. Introduction
Let
be a compact metric space and let
be a continuous map. Then the rule
defines a discrete dynamical system. For an given initial condition
, the set
defines the orbit of the system.
A point
is called periodic if
for some positive integer
, where
(
times). The least such
is called the period of the point
. A map
is called transitive if for any pair of non-empty open sets
in
, there exist a positive integer
such that
. A map
is called weakly mixing if for any pairs of non-empty open sets
and
in
, there exists ![]()
such that
for
. It is known that for any continuous self map
, if
is weakly mixing and
are non-empty open sets, then there exists a
such that
for
. A map
is called mixing or topologically mixing if for each pair of non-empty open sets
in
, there exists a positive integer
such that
for all
.
A map
is called sensitive if there exists a
such that for each
,
there exists
and a positive integer
such that
and
. A map
is called strongly sensitive if there exists a
such that for each
and each neighborhood
of
, there exists a positive integer
such that
for all
. For Details refer [1]-[3].
For non-empty open subsets
of
, Define,
![]()
![]()
![]()
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For any two continuous self maps
on
, define
.
It can be seen that
defines a metric on the space of all continuous functions.
Let
be sequence of continuous maps on
. The sequence is said to converge uniformly to a function
if for each
,
such that
.
In real systems, it is often observed that due to natural constraints, any modeling of a system yields a discrete or continuous system which approximates the behavior of the original system. Thus, it is interesting to see how the dynamics of approximations effect the dynamics of the original system. In this direction, we prove that many of the dynamical properties of the system cannot be concluded even with the strongest form of approximation. We prove that if a sequence of sensitive maps converges uniformly to a function
, the map
need not be sensitive. Similar conclusions can be made for dynamical properties like transitivity and dense set of periodic points. We derive conditions under which uniform limit of weakly mixing/topologically mixing maps is weakly mixing/topologically mixing. In recent times, some of these questions have grabbed attention. In [4], authors claimed that uniform limit of a sequence of transitive maps is transitive. However, the claim proved to be false, was corrected in [5] [6] where authors proved that uniform convergence of transitive maps need not be transitive.
In this paper we try to answer some of the above raised questions. We prove that many of the above mentioned properties are not preserved even under the strongest notion of convergence. In particular we prove that if the maps
have positive topological entropy, the limit map need not have a positive topological entropy. We derive conditions under which properties like transitivity, dense periodic points and sensitive dependence on initial conditions are preserved.
2. Main Results
We first give some examples to show that properties under consideration need not be preserved under uniform convergence.
Example 1: Let
be a sequence of irrational numbers converging to 1. Let
defined as,
![]()
It may be noted that as
are irrationals, each
is transitive but the limit is the identity map which is not transitive.
If we take the sequence
of rational numbers converging to an irrational number
, then the sequence
is a sequence of maps with dense periodic; points but the limit function does not contain any periodic point.
Example 2: Let
be the unit interval and
be the unit circle. Then the product
is a cylinder and the metric
gives the product topology on it.
For each
, define
as,
![]()
Then for the map
, as any two points at different heights rotate with different velocities and move apart in finitely many iterates, for each
and any neighborhood
of
, there are points in
(at different height) which move apart in finitely many iterates and hence each
is strongly sensitive. However, the limit of the sequence
is the identity map which is not sensitive and hence sensitivity/strong sensitivity is not preserved under uniform convergence.
From the examples above, it is proved that the dynamical behavior of a sequence of dynamical systems need not be preserved even under the strongest form of convergence. We now give some necessary and sufficient conditions under which some of the dynamical properties of a sequence of dynamical systems
are preserved in the limit map
.
Result 1: Let
be a compact metric space and let
be a sequence of self maps on
converging uniformly to
. Then,
is transitive if and only if for any pair of non-empty open sets in
,
.
Proof. Let
be transitive and let
be two non-empty open sets in
. As
is transitive, there exists
such that
. Therefore, there exists
such that
. As
, there exists
such that
. Thus,
. Consequently
and proof of the forward part is complete.
Conversely, let
be two non-empty open sets in
. Let
and
. Choose
such that
and
. Let
and
. By given condition, there exists
such that
. Let
. As
uniformly, there exists
such that
. Choose
. Then,
implies that there exist
such that
. Further, as
,
. Thus,
. Thus
. As the proof can be replicated for any pair of non-empty open sets
,
is transitive.
Result 2: Let
be a compact metric space and let
be a sequence of self maps on
converging uniformly to
. Then,
is weakly mixing if and only if for any pair of non-empty open sets
,
in
,
.
Proof. Let
be weakly mixing and let
be two non-empty open sets in
. As
is weakly mixing, there exists
such that
. Therefore, there exists
such that
. As
uniformly, there exists
such that
. Thus,
. Consequently
.
Conversely, let
be non-empty open sets in
. Let
and
for
. Choose
such that
and
. Let
and
. By given condition, there exists
such that
. Let
. As
uniformly, there exists
such that
. Choose
. Then,
implies that there exist
such that
. Further, as
,
. Thus,
. Thus
. As the proof can be replicated for any collection of non-empty open sets in X,
is weakly mixing.
Corollary 1: Let
be a compact metric space and let
be a sequence of self maps on
converging uniformly to
. Then,
is topologically mixing if and only if for any pair of non-empty open
sets
in
,
is cofinite.
Proof. If
is cofinite for a pair of open sets
in
, then there exists a
such that
and
interact under the map
at time instant
, for all
. Using previous result,
. However as
is cofinite, such interaction happens for all
and hence
for all
. Thus
is topologically mixing.
Result 3: Let
be a compact metric space and let
be a sequence of self maps on
converg-
ing uniformly to
. Then,
is sensitive if and only if there exists a
such that for any non-empty open set
in
,
.
Proof. Let
be sensitive with sensitivity constant
and let
be a non-empty open set in
. As
is sensitive, there exists
and
such that
. As
uniformly,
there exists
such that
. Consequently
for all
. Thus
for all
. Consequently,
and proof of the forward part is complete.
Conversely, let
be an arbitrary non-empty open set and let
. As
uniformly, there exists
such that
,
. Choose
. Then, there exists
such that
. Also, as
,
. Consequently,
. As
was arbitrary,
is sensitive.
Corollary 2: Let
be a compact metric space and let
be a sequence of self maps on
converging uniformly to
. Then,
is strongly sensitive if and only if there exists a
such that for
any non-empty open set
in
,
is cofinite.
Proof. Similar
Result 4: Let
be a compact metric space and let
be a sequence of self maps on
converging uniformly to
. Then,
has dense set of periodic points if and only if for any non-empty open
set
in
,
, where
varies over all neighborhoods of
.
Proof. Let
has dense set of periodic points and let
be a non empty open set in
. Let
be periodic with period
. As
uniformly,
converges to
. Thus, for each neighborhood
of
, there exists
such that
for all
. Thus,
for all
which implies
. As the steps can be repeated for any neighborhood
of
,
. Consequently,
.
Conversely, let
be an arbitrary non-empty open set in
and let
. Thus, there exists
such that for each neighborhood
of
,
, i.e. every
neighborhood
of
contains the tail of the sequence
. Thus
converges to
. But as
uniformly,
and
is a periodic point in
. As
was arbitrary,
has a dense set of periodic points.
Acknowledgements
The author would like to thank the referees for their valuable comments and suggestions.