Invariant probabilities for discrete time linear dynamics via thermodynamic formalism

Author(s)

Lopes, Artur O. ^[1] ; Messaoudi, Ali ^[2] ; Stadlbauer, Manuel ^[3] ; Vargas, Victor ^[1]

Affiliation

^[1] IME UFRGS, Porto Alegre, RS - Brazil

^[2] MAT UNESP, Presidente Prudente, SP - Brazil

^[3] IM UFRJ, Rio De Janeiro, RJ - Brazil

• Document type: Journal article
• Source: Nonlinearity; v. 34, n. 12, p. 8359-8391, DEC 2021.
• Web of Science Citations: 0

Abstract

We show the existence of invariant ergodic sigma-additive probability measures with full support on X for a class of linear operators L : X -> X, where L is a weighted shift operator and X either is the Banach space c(0)(R) l(p)(R) 1 <= p < infinity. In order to do so, we adapt ideas from thermodynamic formalism as follows. For a given bounded Holder continuous potential A:X -> R <i , we define a transfer operator L-A X and prove that this operator satisfies a Ruelle-Perron-Frobenius theorem. That is, we show the existence of an eigenfunction for L-A A over bar L-A over bar {*} X with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of LA a priori probability on the kernel of L. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces. (AU)

• FAPESP's process: 13/24541-0 - Ergodic and qualitative theory of dynamical systems
• Grantee: Claudio Aguinaldo Buzzi
• Support Opportunities: Research Projects - Thematic Grants