Communicated by Stephen Smale, April 29, 1970 0. Introduction. Let M be a finite dimensional Riemann manifold without boundary. Kupka [5], Sacker [9], and others have studied perturbations of a flow or diffeomorphism of M leaving invariant a compact submanifold. Anosov [2] considers perturbations of a nonsingular flow, which of course leaves invariant each leaf of the foliation by trajectories. In both problems there is an assumption of hyperbolicity in planes normal to the submanifold, or trajectories.We present a more general theory of diffeomorphisms hyperbolic to a compact laminated subset AC AT. (This includes flows, by considering the time one map.) We suppose A is the disjoint union of injectively immersed submanifolds, called leaves, whose tangent planes vary continuously on A. The diffeomorphism is assumed to permute the leaves, and its differential is more hyperbolic normal to the leaves than tangent to them. The main theorems assert that this situation persists under small perturbations of the diffeomorphisms. By means of the technical device of unwinding the leaves of A, the proofs are reduced to the case of a single invariant, closed submanifold.Applications are made to stability of group actions and fl-stability. See also references and AMS 1969 subject classifications. Primary 3465, 2240, 3451, 3453, 5736, 5482.
Library of Congress Cataloging-in-Publication Data Complexity and real computation / Lenore Blum ... [et al.].p. cm. IncIudes bibliographical references and index.
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Two closely-related pseudo-random sequence generators are presented: The lIP generator, with input P a prime, outputs the quotient digits obtained on dividing by P. The x mod N generator with inputs N, Xo (where N P. Q is a product of distinct primes, each congruent to 3 mod 4, and x 0 is a quadratic residue mod N), outputs bob1 b2" where bi parity (xi) and xi+ x mod N. From short seeds each generator efficiently produces long well-distributed sequences. Moreover, both generators have computationally hard problems at their core. The first generator's sequences, however, are completely predictable (from any small segment of 21PI + consecutive digits one can infer the "seed," P, and continue the sequence backwards and forwards), whereas the second, under a certain intractability assumption, is unpredictable in a precise sense. The second generator has additional interesting properties: from knowledge of Xo and N but not P or Q, one can generate the sequence forwards, but, under the above-mentioned intractability assumption, one can not generate the sequence backwards. From the additional knowledge of P and Q, one can generate the sequence backwards; one can even "jump" about from any point in the sequence to any other. Because of these properties, the x mod N generator promises many interesting applications, e.g., to public-key cryptography. To use these generators in practice, an analysis is needed of various properties of these sequences such as their periods. This analysis is begun here.
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