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We study a given fixed continuous function φ : S 1 → R and an endomorphism f : S 1 → S 1 , whose f -invariant probability measures maximize φ dµ. We prove that the set of endomorphisms having a φ maximizing invariant measure supported on a periodic orbit is C 0 dense. Mathematics Subject Classification: 37E10, 37A99doi:10.1088/0951-7715/21/10/008 fatcat:nc2nkrc7jram5llvhhnnzu2toa