Combinatorics geometry and attractors of quasi quadratic maps

The Milnor problem on one dimensional attractors is solved for S unimodal maps with a non degenerate critical point c It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point This theorem follows from a geometric study of the critical set c of a non renormalizable map It is proven that the scaling factors characterizing the geometry of this set go down to at least exponentially This resolves the problem of the non linearity control in small scales The proofs strongly involve ideas from renormalization theory and holomorphic dynamics x Introduction Let f be an S unimodal map see the de nitions later of the interval with a non degenerate critical point c Let us call such a map quasi quadratic As usual x denotes the limit set of the forward orb x The following theorem solves the Milnor problem M Theorem on the Measure Theoretic Attractor Let f be a quasi quadratic map normalized by the condition f c Then there is a unique set A a measure theoretic attractor in the sense of Milnor such that A x for Lebesgue almost all x and only one of the following three possibilities can occur A is a limit cycle A is a cycle of intervals A is a Feigenbaum like attractor This result gives a clear picture of measurable dynamics for the maps under consid eration Let us explain the words used in the statement A limit cycle is the periodic orbit whose basin of attraction has non empty interior A cycle of intervals is the union of nitely many intervals In with disjoint interiors cyclically interchanged by the dynamics A Feigenbaum like attractor is an invariant Cantor set of the following structure A On where O O is a nested sequence of cycles of intervals of increasing periods It makes sense to compare the above Theorem with its topological counterpart known since late s Theorem on the Topological Attractor MT G JR vS Let f be an S unimodal map normalized by the condition f c Then there is a unique set a topological attractor such that x for a generic x and only one of the following three possibilities can occur Supported in part by NSF grant DMS and a Sloan Research Fellowship i is a limit cycle ii is a cycle of intervals iii is a Feigenbaum like attractor From this point of view the Theorem on the Measure Theoretic Attractor says that the map f has a unique measure theoretic attractor A coinciding with the topological attractor In cases i and iii this was proven by Guckenheimer G In case ii it is known that p