**MSC:**- 30F25 Ideal boundary theory
- 34C35 Dynamical systems, See also {54H20, 58Fxx, 70-XX}

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 $P^2$.