1. IntroductionLet 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 ResultsWe 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,
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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,
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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.