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We extended the study of the linear fractional self maps (e.g. byCowen-MacCluer and Bisi-Bracci on the unit balls) to a much more general classof domains, called generalized type-I domains, which includes in particular theclassical bounded symmetric domains of type-I and the generalized balls. Sincethe linear fractional maps on the unit balls are simply the restrictions of thelinear maps of the ambient projective space (in which the unit ball isembedded) on a Euclidean chart with inhomogeneous coordinates, and in thisarticle we always worked with homogeneous coordinates, here the term linear mapwas used in this more general context. After establishing the fundamentalresult which essentially says that almost every linear self map of ageneralized type-I domain can be represented by a matrix satisfying the"expansion property" with respect to some indefinite Hermitian form, we gave avariety of results for the linear self maps on the generalized balls, such asthe holomorphic extension across the boundary, the normal form and partialdouble transitivity on the boundary for automorphisms, the existence and thebehavior of the fixed points, etc. Our results generalize a number of knownstatements for the unit balls, including, for example, a theorem of Bisi-Braccisaying that any linear fractional map of the unit ball with more than twoboundary fixed points must have an interior fixed point.

Generalizations of linear fractional maps for classical symmetric domains and related fixed point theorems for generalized balls