For a degree-q self-map of P^n, the number of totally invariant codimension-r subvarieties is at most (n+1 choose r), and the bound is optimal.
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Sharp Bounds for Totally Invariant Cycles of Projective Varieties
For a degree-q self-map of P^n, the number of totally invariant codimension-r subvarieties is at most (n+1 choose r), and the bound is optimal.