Stefan-M. Heinemann
\boldmath Shilov Boundary, Dynamics and Entropy in $C^2$\unboldmath
Preprint series: Mathematica Gottingensis
MSC:
30F25 Ideal boundary theory
34C35 Dynamical systems, See also {54H20, 58Fxx, 70-XX}
ZDM: I60
Abstract: For domains in $C^2$ which are defined by consideration of
the dynamics of a holomorphic endomorphism $T:C^2\to C^2$ we
investigate the Shilov boundary $\partial_{SH}K(T)$ of
their closure $K(T)$. We show that the complement of the
Shilov boundary in the topological boundary $\partial K(T)$
foliates into complex analytic sets. Moreover, the Shilov
boundary is identified as the Julia set $J(T)$ of the
defining endomorphism, equals the closure of the set of
repelling periodic points of $T$ and also the support of the
unique measure of maximal entropy of $T$. We do neither need
any assumption on the smoothness of the boundary of $K(T)$
nor that $T$ extends to the two-dimensional complex projective
space $P^2$.
Keywords: Holomorphic Dynamics, Shilov Boundary, Entropy, Julia Set