TAILIEUCHUNG - Đề tài " Decay of geometry for unimodal maps: An elementary proof "

We prove that a nonrenormalizable smooth unimodal interval map with critical order between 1 and 2 displays decay of geometry, by an elementary and purely “real” argument. This completes a “real” approach to Milnor’s attractor problem for smooth unimodal maps with critical order not greater than 2. 1. Introduction The dynamical properties of unimodal interval maps have been extensively studied recently. A major breakthrough is a complete solution of Milnor’s attractor problem for smooth unimodal maps with quadratic critical points. . | Annals of Mathematics Decay of geometry for unimodal maps An elementary proof By Weixiao Shen Annals of Mathematics 163 2006 383 404 Decay of geometry for unimodal maps An elementary proof By Weixiao Shen Abstract We prove that a nonrenormalizable smooth unimodal interval map with critical order between 1 and 2 displays decay of geometry by an elementary and purely real argument. This completes a real approach to Milnor s attractor problem for smooth unimodal maps with critical order not greater than 2. 1. Introduction The dynamical properties of unimodal interval maps have been extensively studied recently. A major breakthrough is a complete solution of Milnor s attractor problem for smooth unimodal maps with quadratic critical points. Let f be a unimodal map. Following 19 let us define a minimal measure-theoretical attractor to be an invariant compact set A such that x w x c A has positive Lebesgue measure but no invariant compact proper subset of A has this property. Similarly we define a topological attractor by replacing has positive Lebesgue measure with is a residual set . By a wild attractor we mean a measure-theoretical attractor which fails to be a topological one. In 19 Milnor asked if wild attractors can exist. For smooth unimodal maps with nonflat critical points this problem was reduced to the case that f is a nonrenormalizable map with a nonperiodic recurrent critical point by a purely real argument. Furthermore in 8 12 it was shown that such a map f does not have a wild attractor if it displays decay of geometry. A smooth unimodal map f with critical order Í sufficiently large may have a wild attractor. See 2 . But in the case h 2 it was expected that f would have the decay of geometry property and thus have no wild attractor this has been verified in the case h 2 so far. In fact in 8 12 it was proved that for S-unimodal maps with critical order Í 2 the decay of geometry property follows from a starting condition . Kozlovski 11 allowed one to get .

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