Invariant curves as solutions of functional equations

摘要: Numerous evolution processes observed in nature are modellized by discrete dynamic systems, which can be interpreted as recurrences or point mappings. One of the simpler R 2→R 2 examples is (see a.o. [3], [5], [7], [8]) $$T: \left( {x,y} \right) \to {X = g\left( \right),\quad Y f\left( \right)} \right)$$ (1) where f,g smooth functions their arguments. The singularities (1) necessarily dimension zero (fixed points and cycles a finite order k≥1) one (invariant curves analoguous invariant manifolds). Let w(x,y) const. α equation an curve (α fixed), then automorphic function satisfying functional for example [1], [3]) $$w\left( {g\left( \right),f\left( w\left( \right)$$ (2) Many mappings form known to possess isolated closed curve. Although most them noninvertible endomorphisms (examples: [4], [7]), some happen at least local diffeomorphisms (example: [8]).