The purpose of this talk is to present an (apparently) new way to look at the intersection complex of a singular variety over a finite field, or, more generally, at the intermediate extension functor on pure perverse sheaves, and an application of this to the cohomology of noncompact Shimura varieties.

pdf/proceedings-of-the-international-congress-of-mathematicians-32epad3tqi.pdf

Proceedings of the International Congress of Mathematicians

V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report is my response. Arnold's invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay. I have listed 18 problems, chosen with these criteria:

Mathematical problems for the next century

This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The equivalence classes of maps which display sensitivity to initial conditions for large sets of initial conditions are characterized.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103907294

Sensitive dependence to initial conditions for one dimensional maps

Generic hyperbolicity in the logistic family

Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of infinity. We call a C1 endomorphism of the compact interval (or the circle) hyperbolic if it has finitely many hyperbolic attracting periodic points and the complement of the basin of attraction of these points is a hyperbolic set. By a theorem of Mañé for C2 maps, this is equivalent to the following conditions: all periodic points are hyperbolic and all critical points converge to periodic attractors. Note that the space of hyperbolic maps is an open subset in the space of real polynomials of fixed degree, and that every hyperbolic map satisfying the mild “no-cycle” condition (which states that orbits of critical points are disjoint) is structurally stable; see [dMvS93]. Theorem 1 solves the 2nd part of Smale’s eleventh problem for the 21st century [Sma00]:

Density of hyperbolicity in dimension one

In this article is constructed the set , open and everywhere dense in , of -stable mappings. is totally disconnected and is topologically conjugate to the topological Markov chain with a finite number of states; for we have and topologically conjugate to . For there exists a hyperbolic structure on .Diagrams: 1Bibliography: 9 items

On Smooth Mappings of the Circle Into Itself

For an arbitrary non-renormalizable unimodal map of the interval, f: I → I, with negative Schwarzian derivative, we construct a related map F defined on a countable union of intervals Δ. For each interval Δ, F restricted to Δ is a diffeomorphism which coincides with some iterate of f and whose range is a fixed subinterval of I. If F satisfies conditions of the Folklore Theorem, we call f expansion inducing. Let c be a critical point of f. For f satisfying f″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds for f: the ω-limit set of Lebesgue almost every point is the interval [f2, f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

Metric properties of non-renormalizable S -unimodal maps. Part I: Induced expansion and invariant measures

In this chapter we shall consider one-dimensional mappings which have been intensely studied during the last few years, from the point of view of both dynamical systems and ergodic theory. Phase spaces of these systems are intervals I ⊂ R 1 and transformations are real-valued functions determined on I and taking their values in I. We shall study invariant measures of one-dimensional maps, especially absolutely continuous invariant measures. The topological aspect of the problem will be discussed in one of the further volumes.

Ergodic Theory of One-Dimensional Mappings

We prove the existence of Sinai-Ruelle-Bowen measures for a class of C2 self-mappings of a rectangle with unbounded derivatives. The results can be regarded as a generalization of a well-known one dimensional Folklore Theorem on the existence of absolutely continuous invariant measures. In an earlier paper [8] analogous results were stated and the proofs were sketched for the case of invertible systems. Here we give complete proofs in the more general case of noninvertible systems, and, in particular, develop the theory of stable and unstable manifolds for maps with unbounded derivatives.

pdf/asymptotic-measures-for-hyperbolic-piecewise-smooth-mappings-4tigycv367.pdf

Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle

We consider the two-parameter family of maps on the circle 



and we find a set M = {(q,ω)} of positive Lebesgue measure such that (q,ω) ∈ M implies the stochastic behavior of fq,ω. We present analytical and numerical results which describe the structure of M as follows.

Absolutely Continuous Invariant Measures for Some Maps of the Circle

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