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We consider a smooth semiflow strongly focusing monotone with respect to acone of rank k on a Banach space. We obtain its generic dynamics, that is,semiorbits with initial data from an open and dense subset of any bounded openset are either pseudo-ordered or convergent to an equilibrium. For the casek=1, it is the celebrated Hirsch's Generic Convergence Theorem. For the casek=2, we obtain the generic Poincare-Bendixson Theorem.

Semiflows strongly focusing monotone with respect to high-rank cones. I: Generic dynamics