On metric properties of unconventional limit sets of contractive non-Archimedean dynamical systems - Mukhamedov, Farrukh M.; Khakimov, Otabek Kh (Taylor and Francis Ltd., 2016)
In this paper, we define the limit set Λξ of an unconventional set of contractive functions {fk} on the unit ball of non-Archimedean algebra. Then, we prove that Λξ is compact, perfect and uniformly disconnected. It is shown that there is a new collection of contractive mappings (Formula presented.) defined on Λξ. Moreover, we establish that the set Λξ coincides with the limit set generated by the semi-group of (Formula presented.). This result allows us to further investigate the structure of Λξ by means of this limiting set. As an application, we demonstrate the existence of invariant measures on Λξ. We should stress that the non-Archimedeanity of the space is essentially used in the paper. Therefore, the methods applied in this paper are not longer valid in the Archimedean setting (i.e. in case of real or complex numbers).