The sets of points with bounded orbits for generalized Chebyshev mappings

We study the dynamical systems given by generalized Chebyshev mappings F-c( z) = z(2) - c (z) over bar (e epsilon R) and show that (1) the set of point s with bounded orbits of F-c(z) is connected and its complement in C boolea n OR {infinity} is simply connected if and only if -4 less than or equal to c less than or equal to 2; (2) if c > 2, then the set of points with bound ed orbits of F-c(z) is Canter set. These results are the analogue of the theory of filled Julia sets of quadra tic polynomials in one complex variable. We show that the mapping F-c(z) ha s relation to an important holomorphic map on the complex projective space P-2.