By Zbigniew Nitecki (auth.), A. Katok (eds.)
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Additional resources for Ergodic Theory and Dynamical Systems II: Proceedings Special Year, Maryland 1979–80
Jk, k > for M. , i < '" l. interior to all under oJ Any ak £ ° and another, > k ~. and hence some high image of a k = fk(c' ) has one endpoint Tk-neighborhood Tk-neighborhood of On the other hand, itself. 2 i-I J bk , of Thus, some image of J is an remains in Ml.. 17) may really fail to be injective. , l. the coding of by a K. l. topological Markov chain separates periodic (and hence eventually periodic) points. However, even in this situation, it is conceivable that some infinite orbit in the topological Markov chain corresponds to a pair of infinite "parallel" orbits.
46 1. For each i < 00 a) Fi is finite b) ni has a finite decomposition ••• un. ~n into disjoint closed sets permuted cyclically by f. Each is either (i) the set of fixedpoints of f2n on an interval where ~ is monotone or (ii) a set with a dense ~In .. ~J ~n orbit. In this case, either ~nln~. (k is topologically mixing or topologically mixing, where ~J n .. = n: . ~J ~J \J = 1, 2) n2 .. ~J are sets meeting at a unique point and interchanged under 2. ni , If there are infinitely many then noo f ~ is f.
2b) U(p, f, S) = Of course, k~O fk(F) = F U(fk(p), f n , fk(S» for any k. 2b) there are two possibilities: either U(p, f n , R) = U(p, f n , L), choice of and fk(S) fk(L) can be made arbitrarily, or else the choices are distinct for any fk+l(s)-neighborhood of k. f[N(f (p), £, f fk+l(p). 2c) in which case every k (S»] is contained in an In any case, we always have U(p, f, F) =U(p, f,R)UU(p, f, L). When fication. 2) needs a slight modi- respects side S at a fixedpoint if the image of every sufficiently small S-neighborhood of an S-neighborhood of p neighborhood collapses to p is contained in (this includes the possibility that some Sp); g flips side S if the image of every 20 sufficiently small S-neighborhood of where T ~ p is a T-neighborhood of A map which flips both sides exchanges sides at S.
Ergodic Theory and Dynamical Systems II: Proceedings Special Year, Maryland 1979–80 by Zbigniew Nitecki (auth.), A. Katok (eds.)