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Ruitenburg's Theorem says that every endomorphism f of a finitely generatedfree Heyting algebra is ultimately periodic if f fixes all the generators butone. More precisely, there is N $\ge$ 0 such that f N +2 = f N , thus theperiod equals 2. We give a semantic proof of this theorem, using dualitytechniques and bounded bisimulation ranks. By the same techniques, we tackleinvestigation of arbitrary endomorphisms between free algebras. We show thatthey are not, in general, ultimately periodic. Yet, when they are (e.g. in thecase of locally finite subvarieties), the period can be explicitly bounded asfunction of the cardinality of the set of generators.

Free Heyting Algebra Endomorphisms: Ruitenburg's Theorem and Beyond