Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems

Abstract

We describe a general method of studying prevalent properties of diffeomorphisms of a compact manifold M, where by prevalent we mean true for Lebesgue almost every parameter ε in a generic finite-parameter family {fε} of diffeomorphisms on M. Usually a dynamical property P can be formulated in terms of properties Pn of trajectories of finite length n. Let P be such a dynamical property that can be expressed in terms of only periodic trajectories. The first idea of the method is to discretize M and split the set of all possible periodic trajectories of length n for the entire family {fε} into a finite number of approximating periodic pseudotrajectories. Then for each such pseudotrajectory, we estimate the measure of parameters for which it fails Pn. This bounds the total parameter measure for which Pn fails by a finite sum over the periodic pseudotrajectories of length n. Application of Newton interpolation polynomials to estimate the measure of parameters that fail Pn for a given periodic pseudotrajectory of length n is the second idea. We outline application of these ideas to two quite different problems:

• Growth of number of periodic points for prevalent diffeomorphisms (Kaloshin–Hunt).
• Palis’ conjecture on finititude of number of “localized” sinks for prevalent surface diffeomorphisms (Gorodetski–Kaloshin).