By Zbigniew Nitecki (auth.), A. Katok (eds.)
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Extra resources for Ergodic Theory and Dynamical Systems II: Proceedings Special Year, Maryland 1979–80
Exist maps f: I .... * There are some partial results on this problem. Jakobson (Ja 1), Szlenk (Sz 1, see also Mi 5), and Misiurewicz and I (see Mi 5) have results that imply that a non-trivial homterval of a a critical point in its w-limit set. c2 map must contain Misiurewicz (Mi 5) and Guckenheimer (G 4) have ruled out non-trivial homtervals for c 3 maps f : I .... I satisfying: (i) (ii) f For has a single critical point: x # c, f Df(c) = 0 has negative Schwartzion derivative: Df(x) # 0 implies 322 2 Df (x) D f (x) < 3 [D f (x) 1 • *I have just learned from R.
7) Remark: (p, S) £ a. For i < 00, if to decompose p E We will use a variant ~(f). [Mi' Mi +l ], then Per(f) " S = R or L. for 1. f. 7: Since M. 1. and are intervals, each component of a nontrivial interval. for S (p, S) or L. R £ a j Thus, Since with j 1. dieting the hypothesis. If j i, > then A(p, S) p E C Mi and so A(a i ) C Mi + l , contra- • The converse is true with the exception that an endpoint might belong to or if one side (p, L) 'I (p, R), (Xi' S of is contains an S-neighborhood of is invariant, M.
On the other hand, generically the sets equal. 4) Theorem (Ja l): A c2 -dense, open set of maps of the circle and 2 interval satisfy Axiom A and are C -structurally stable. 26, but predating it), Cr genericity of >l(f) = Per (f) for r > 1. 5) of Theorem (Ja 1, Y 1, CMN): Cr maps f:I-+I, Everyelement for every of a Cr-residual set satisfies > 0, r f 0, (f) = Per (f) • I would like to mention a result concerning the return-times of a neighborhood of x E 0,(f) to itself. recurrent (resp. periodic) point of It is known in general that a f is also recurrent (resp.