Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains - Mellon, P. (Mathematica Scandinavica, 2015)
Let g be a fixed-point free biholomorphic self-map of a bounded symmetric domain B. It is known that the sequence of iterates (g^n) may not always converge locally uniformly on B even, for example, if B is an infinite dimensional Hilbert ball. However, g=g_a° T, for a linear isometry T, a=g(0) and a transvection g_a, and we show that it is possible to determine the dynamics of g_a. We prove that the sequence of iterates (g_a^n) converges locally uniformly on B if, and only if, a is regular, in which case, the limit is a holomorphic map of B onto a boundary component (surprisingly though, generally not the boundary component of a\|a\|). We prove (g_a^n) converges to a constant for all non-zero a if, and only if, B is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.