Yosef Yomdin

Volume growth and entropy

An inequality is proved, bounding the growth rates of the volumes of iterates of smooth submanifolds in terms of the topological entropy. ForCx-smooth mappings this inequality implies the entropy conjecture, and, together with the opposite inequality, obtained by S. Newhouse, proves the coincidence of the growth rate of volumes and the topological entropy, as well as the upper semicontinuity of the entropy.

Tame geometry with application in smooth analysis

Preface.- Introduction and Content.- Entropy.- Multidimensional Variations.- Semialgebraic and Tame Sets.- Some Exterior Algebra.- Behavior of Variations under Polynomial Mappings.- Quantitative Transversality and Cuspidal Values for Polynomial Mappings.- Mappings of Finite Smoothness.- Some Applications and Related Topics.- Glossary.- References.

C k-resolution of semialgebraic mappings. Addendum toVolume growth and entropy

We prove that a bounded semialgebraic function can be (piecewise) reparametrized in such a way that all the derivatives up to a fixed orderk, with respect to new coordinates, are small, and the number of pieces is effectively bounded.

Cauchy Type Integrals of Algebraic Functions

AbstractWe consider Cauchy-type integrals

OSCILLATION OF ANALYTIC CURVES

I(t) = \frac{1}{{2\pi i}} \int_\gamma {\frac{{g(z)dz}}{{z - t}}}

Accuracy of spike-train Fourier reconstruction for colliding nodes

withg(z) an algebraic function. The main goal is to give constructive (at least, in principle) conditions forI(t) to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the monodromy group of the algebraic functiong, the geometry of the integration curve γ, and the analytic properties of the Cauchy-type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy-type integrals of algebraic functions appear in the infinitesimal versions of two classical open questions in Analytic Theory of Differential Equations: the Poincaré Center-Focus problem and the second part of Hilbert’s 16-th problem.

Flexible high-order discretization of geometric data for global motion planning

In this paper we consider several nonlinear systems of algebraic equations which can be called “Prony-type.” These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of “maximal possible accuracy” of solving Prony-type systems, putting stress on the “local” behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that “global” solution techniques such as Pronys algorithm and the ESPRIT method are suboptimal when compared to this theoretical “best local” behavior.

Semialgebraic geometry of polynomial control problems

The number of zeroes of the restriction of a given polynomial to the trajectory of a polynomial vector field in (Cn, 0), in a neighborhood of the origin, is bounded in terms of the degrees of the polynomials involved. In fact, we bound the number of zeroes, in a neighborhood of the origin, of the restriction to the given analytic curve in (Cn, 0) of an analytic function, linearly depending on parameters, through the stabilization time of the sequence of zero subspaces of Taylor coefficients of the composed series (which are linear forms in the parameters). Then a recent result of Gabrielov on multiplicities of the restrictions of polynomials to the trajectories of polynomial vector fields is used to bound the above stabilization moment. Introduction Let ψ : (C, 0) → (C, 0) be an analytic curve in C. We assume that an analytic mapping ψ = (ψ1, . . . , ψn) is regular in a neighborhood of 0 ∈ C. It is known that for any compact analytic family of analytic functions Qλ : C → C, the number of isolated zeroes of Qλ on ψ near the origin is uniformly bounded in λ (see [5]). In particular, this is true for Qλ—all the polynomials of degree d. However, in many situations it is important to have an effectively computable bound on the number of zeroes. Such a bound cannot be produced by the approach of [5]. Some additional approaches to this problem have been developed: Khovanski’s theory of Pfaffian functions ([10]) allows one to effectively bound the number of real zeroes for solutions of some special type of differential equations. Recently, serious progress has been achieved in extending these bounds to non-Pfaffian situations and to complex zeroes ([6], [7]). Another approach, based on Bernstein-type inequalities and their relation to the number of zeroes, has been proposed in [13], and developed for linear differential equations in [9]. However, even for ψ(t) the trajectory of a polynomial vector field and Qλ-polynomials of degree d, a serious gap remains. The results of [6] (see also [11]) bound effectively only the multiplicities of zeroes of Qλ/ψ, but not their number. On the other hand, the results of [9] cannot be extended directly from one differential equation to a system. Let us stress that the main difficulty in the problem above concerns the situation where Qλ is a small perturbation of Qλ0 , for which Qλ0/ψ ≡ 0. The number of Received by the editors January 4, 1996. 1991 Mathematics Subject Classification. Primary 30B10, 34A20, 30C55, 34A25, 34C15. This research was partially supported by the Israel Science Foundation, Grant No. 101/95-1, and by the Minerva Foundation. c ©1998 American Mathematical Society