This is an announcement for the paper "On asymptotic transitivity in Banach spaces" by J. Talponen. Abstract: We introduce a flexible almost isometric version of the almost transitivity property of Banach spaces. With the help of this new notion we generalize to several directions a strong recent rotational characterization of Hilbert spaces due to Randrianantoanina. This chracterization is a partial answer to the classical Banach-Mazur rotation problem. Archive classification: Functional Analysis Mathematics Subject Classification: 46C15; 46B04 Remarks: 15 pages The source file(s), asyams.tex: 58021 bytes, is(are) stored in gzipped form as 0610547.gz with size 17kb. The corresponding postcript file has gzipped size 82kb. Submitted from: talponen@cc.helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610547 or http://arXiv.org/abs/math.FA/0610547 or by email in unzipped form by transmitting an empty message with subject line uget 0610547 or in gzipped form by using subject line get 0610547 to: math@arXiv.org.