Wednesday, 07 November 2018
Helge Glöckner: Dynamical systems on totally disconnected groups
In the structure theory of totally disconnected, locally compact groups started in [6], the iteration of automorphisms plays an essential role. Several subgroups can be associated to an automorphism \(f\) of a locally compact group \(G\), like the contraction group \(con(f)\) of all group elements whose forward orbit under \(f\) converges to the neutral element \(e\), or the group \(par(f)\) of all group elements whose forward orbit is relatively compact. For totally disconnected \(G\), the study of such subgroups was started in [1], and connections were established there to other notions from the structure theory of totally disconnected groups (like tidy subgroups and the scale). In the talk, I'll give an introduction to this area of research, including some recent results both in the general case and for the special case of automorphisms (and endomorphisms) of Lie groups over local fields (as in [2]-[5]).Bibliography:
[1] U. Baumgartner and G. Willis, Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Isr. J. Math. 142 (2004), 221-248.
[2] T.P. Bywaters, H. Glöckner, and S. Tornier, Contraction groups and passage to subgroups and quotients for endomorphisms of totally disconnected locally compact groups, Isr. J. Math. 227 (2018), 691-752.
[3] H. Glöckner, Invariant manifolds for analytic dynamical systems over ultrametric fields, Expo. Math. 31 (2013), 116-150.
[4] H. Glöckner, Endomorphisms of Lie groups over local fields, pp. 101-165 in: D.R. Wood, J. de Gier, C.E. Praeger, and T. Tao (eds.) "2016 MATRIX Annals," Springer-Verlag, 2018.
[5] H. Glöckner and G.A. Willis, Classification of the simple factors appearing in composition series of totally disconnected contraction groups, J. Reine Angew. Math. 634 (2010), 141-169.
[6] G.A. Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), 341-363.