2024/12/09 by Maslyuchenko, Oleksandr V., Morawiec, Janusz, Zürcher, Thomas
#28D05 #39B12 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 37E05 #Secondary 26A30
paper · doi:10.48550/arxiv.2412.06406
It is well-known that the Lebesgue measure is the unique absolutely continuous invariant probability measure under the p-adic transformation. The purpose of this paper is to characterize the family of all invariant probability measures under the p-adic transformation and to provide some description of them. In particular, we describe the subfamily of all atomic invariant measures under the p-adic transformation as well as the subfamily of all continuous and singular invariant probability measures under the p-adic transformation. Iterative functional equations play the base role in our considerations.