Invariant measures from locally bounded orbits

Abstract

Motivated by recent investigations of Sophie Grivaux and Étienne Matheron on the existence of invariant measures in Linear Dynamics, we introduce the concept of locally bounded orbit for a continuous linear operator $T:X⟶X$ acting on a Fréchet space $X$, and we use this new notion to construct (non-trivial) $T$-invariant probability Borel measures on $(X,\mathcal{B}(X))$.

arXiv comment22 pages

References in corpus (7)

• Recurrence properties of hypercyclic operators
• Recurrent Linear Operators
• Frequently recurrent operators
• Transitive points via Furstenberg family
• Strong transitivity properties for operators
• Recurrent subspaces in Banach spaces
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Cited by in corpus (4)

• Two remarks on the set of recurrent vectors
• Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems
• Shifts on trees versus classical shifts in chain recurrence
• Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree