Dynamical properties of weighted composition operators on the space of smooth functions

Dynamical properties of weighted translation operators on the Schwartz space $$\mathcal {S}(\mathbb {R})$$S(R)

“…Together with(11) and(12) this proves the inductive hypothesis. (iii) ⇒ (i) By Theorem 1 the operator T w is hypercyclic and therefore we only need to show that T w has a dense set of periodic points.…”

“…The next theorem characterizes the class of mixing weighted translation operators on S(R). In Example 3 we will show that this class is strictly smaller than the class of hypercyclic operators (note that for weighted composition operators acting on the space of smooth functions on the real line those classes are equal, see [11]).…”

“…In the first theorem we show that hypercyclicity of these operators is equivalent to weak mixing. This is not surprising since for many natural operators the situation is the same (see [11] for weighted composition operators on the space of smooth functions, see [10] for weighted composition operators on Banach spaces of continuous and integrable functions, see [12] and [7] for weighted backward shifts and weighted bilateral shifts on sequence spaces). Let us emphasize that there are hypercyclic operators which are not weakly mixing (see [5]).…”

“…is hypercyclic (see [2]). In fact Birkhoff's operator is an example of a composition operator and the dynamics of this kind of operators was later on studied by various authors (see [8] for composition operators on spaces of holomorphic functions, see [3] for composition operators on spaces of real analytic functions, see [10] for weighted composition operators on Banach spaces of continuous functions and L p spaces, see [11] for weighted composition operators on the space of smooth functions). Surprisingly, as shown below, composition operators are never hypercyclic when acting on the space S(R).…”

Topological transitivity and mixing of composition operators

“…Primary 47A16, 47B33; Secondary 37A25 . 1 Partially supported by São Paulo Research Foundation (FAPESP) grant # 2015/20731- 5. the literature for regular functions, like holomorphic functions (see for instance [14]) or smooth functions (see [12]). Here, we focus on measurable functions and L p -spaces.…”