Residual windows for partial semigroups and graph-directed conservative dynamics

We prove a residual-window transitivity theorem for countable families of partial homeomorphisms. Each generator is given by an open structural homeomorphism, while its admissible domain is only assumed to be comeagre in the structural domain and may have empty interior. If the structural partial semigroup is open-set transitive, then the admissible system is open-set transitive in the branch sense. More precisely, there is a forward-invariant comeagre set of initial points such that, from each of its points, admissible branches can hit any prescribed sequence of nonempty open targets. The dense-open window theorem is recovered as a special case. Combined with the generic transitivity theorem of Koropecki and Nassiri, the result gives residual-window statements for generic conservative surface pairs and finite tuples. A graph-directed formulation gives analogous conclusions, including a private-edge criterion for strongly connected conservative graphs. The hypothesis of comeagreness is sharp: density and full measure do not suffice.

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