Abstract
In this paper, recurrent C 0 -semigroups are introduced and investigated. It is proved that, despite hypercyclic C 0 -semigroups, recurrent C 0 -semigroups can be found on finite-dimensional Banach spaces. Some criteria are stated for recurrence, which is based on open sets, neighborhoods of zero, and special eigenvectors. It is established that having a dense set of recurrent vectors is a sufficient and necessary condition for a C 0 -semigroup to be recurrent. Moreover, the direct sum of recurrent C 0 -semigroups is investigated.
Highlights
A hypercyclic operator T on X, with a dense set of periodic points is named a chaotic operator
One of the significant structures that are considered by mathematicians is C0-semigroups
A C0-semigroup (Tt)t≥0 on X is named hypercyclic if it has a dense orbit or equivalently for any open sets U and V of X, there is t > 0 such that T−t 1(U) ∩ V ≠ φ
Summary
We begin this section by defining the concept of recurrent C0-semigroup. Definition 3. We say a C0-semigroup (Tt)t≥0 is recurrent if for any open and nonempty set U, some t > 0 can be found such that T−t 1(U) ∩ U ≠ φ. E following lemma states that, for a C0-semigroup (Tt)t≥0 and any open and nonempty set U, T−t 1(U) ∩ U ≠ φ for infinitely many t > 0. Since (St)t≥0 has a dense set of recurrent vectors, there exists x ∈ Rec(St)t≥0 such that x ∈ φ− 1(U). E proof is evident by eorem 5 and this fact that hypercyclic C0-semigroups do not exist on finite-dimensional spaces. A C0-semigroup is constructed with a dense set of frequently recurrent vectors. (Tt)t≥0 is a chaotic C0-semigroup (Example 7.10 in [4]) It has a dense set of periodic points in X. erefore,. (Tt)t≥0 has a dense set of frequently recurrent vectors
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