No abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Our goal in this paper is to establish for flows results analogous to those obtained for discrete dynamical systems in [5].To formulate the main result, we begin by recalling from [4] the notion of a filtration for a flow. Let 4 be the flow on compact n-manifold M generated by a C' vector field X = 4 Definition. A filtration for 4 (or X) is a finite sequence '1 = { M0.. ., Mk } of compact submanifolds with boundary such that (i) 0=MOCMIC-CMk=M (ii) dim Mi = n (V i > 1) (iii) ot [Mi] cintMi (Vt>O) (iv) The flow is transverse to the boundary of each Mi-that is, for x E aMi (O< i < k), X. is not tangent to aMi Given a filtration 'DTh, the maximal 4-invariant subset of Mi-Mi is denoted Ki (DI)= n { 4t[ Mi -intMi I]: :t E=R) and we let K (9t)= U { Ki ( D): i = 1.., k}. We shall call the filtration fine if K (9Dth) = i2(0) (the set of non-wandering points for 4). If 4 does not admit a fine filtration, we will look for a sequence 6Yi of filtrations for which K (, (DT) tend to 52 (p). More precisely, Definition. A filtration GM = {M0o, Mj,..., M1 } refines the filtration 'X = { M,,..., Mk} if for each a = 1..., 1 there exists ia' 1 < io < k, such that Received May 18, 1973. *Alfred P. Sloan Foundation Fellow. All use subject to JSTOR Terms and Conditions 1030 Z. NITECKI AND M. SHUB. Definition. A sequence 912, ... of filtrations for 4 is said to be fine if (i) ?Tih+I refines 9iT (i= 1,...) (ii) n K ((

The well-known connection between braids and mapping classes on the n-punctured disc can be exploited to decide, using braid-theoretic techniques, the place of a given isotopy class in the Nielsen-Thurston classification as reducible or irreducible, and in the latter case, as periodic or pseudo-Anosov.

No abstract

Abstract. We show that, for maps of the interval, the non-wandering set of the map coincides with the non-wandering set of each of its odd powers, while the nonwandering set of any of its even powers can be strictly smaller.