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

### Anton Gorodetski

University of California, Irvine, USA### Brian Hunt

University of Maryland, College Park, USA### Vadim Kaloshin

University of Maryland, College Park, United States

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## 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

*P**n*of trajectories of ﬁnite length

*n*. Let

**be such a dynamical property that can be expressed in terms of only periodic trajectories. The ﬁrst idea of the method is to discretize**

*P**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

*P**n*. This bounds the total parameter measure for which

*P**n*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

*P**n*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).