Michael Shub

A simple unpredictable pseudo random number generator

Two closely-related pseudo-random sequence generators are presented: The

Complexity of Bezout’s Theorem II Volumes and Probabilities

{1 / P}

Neighborhoods of hyperbolic sets

generator, with input P a prime, outputs the quotient digits obtained on dividing 1 by P. The

Complexity of Bezout's theorem IV: probability of success; extensions

x^2 \bmod N

Complexity of Bezout's theorem V: polynomial time

generato...

Stably Ergodic Dynamical Systems and Partial Hyperbolicity

In this paper we study volume estimates in the space of systems of n homegeneous polynomial equations of fixed degrees d i with respect to a natural Hermitian structure on the space of such systems invariant under the action of the unitary group. We show that the average number of real roots of real systems is D 1/2 where D = Π d i is the Be zout number. We estimate the volume of the subspace of badly conditioned problems and show that volume is bounded by a small degree polynomial in n, N and D times the reciprocal of the condition number to the fourth power. Here N is the dimension of the space of systems.

A remark on the Lefschetz fixed point formula for differentiable maps

Inventiones math. 9, 121 - 134 (1970) Neighborhoods of Hyperbolic Sets M. HIRSCH, J. PALIS, C. PUGH, and M. SHUB (Univ. of Warwick) w 1. Introduction In this paper we study the asymptotic behavior of points near a compact hyperbolic set of a C r diffeomorphism (r__>l)f: M - ~ M , M being a compact manifold. The purpose of our study is to complete the proof of Smales O-stability Theorem by demonstrating (2.1), (2.4) of [6]. O denotes the set of non-wandering points for f Smales Axiom A requires [5]: (a) O has a hyperbolic structure, (b) the periodic points are dense in O. Hyperbolic structure, the stable manifold of O, and fundamental neighborhoods are discussed in ~j 2 and 5. The result of [-6] proved here is: I f f obeys Axiom A then there exists a proper fundamental neighbor- hood V for the stable manifold of f2 such that the union of the unstable manifold oft2 and the forward orbit of V contains a neighborhood of 0 in M. As a consequence we have: l f f obeys Axiom A then any point whose orbit stays near 0 is asymp- totic with a point of 0. Section 8 of the mimeographed version of [ 1 ] contains a generalization of the above results with an incorrect proof. A correct generalization is: (1.1) Theorem. I f A is a compact hyperbolic set then WU(A)uO+ V contains a neighborhood U of A, where V is any fundamental neighbor- hood for WS(A) and 0+ V= U f ( v ) . i f A has local product structure n>O then a proper fundamental neighborhood may be found and any point whose forward orbit lies in U is asymptotic with some point of A. Theorem (1.1) is proved in w 5, local product structure is discussed in [-5] and in w In w we prove the analogous theorems for flows. Here is an example, due to Bowen, of a compact hyperbolic set A which does not have local product structure, has no proper fundamental neighborhood and for which there are points asymptotic to A without being asymptotic with any point of A.

Computational complexity : on the geometry of polynomials and a theory of cost: II

We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in

Expanding endomorphisms of the circle revisited

n + 1

Ergodicity of Anosov actions

complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed.