---
tag: G14BVRT221
kind: problem
name: Smale's problems on dynamics
msc: [37C20, "37E05", 37D45]
uses: [E1Z444F6TV, EY2W34JZA4, T75CZ5H47N]
about: [116TPXR5FS]
sources:
  - url: https://en.wikipedia.org/wiki/Smale%27s_problems
history:
  posed_by: "Stephen Smale"
  first_stated: "1998"
---

## Problem

In 1998 Smale published a list of eighteen problems for the twenty-first century.
The following problems of the list concern dynamical systems.

1. Problem 10, the $C^r$ closing lemma. Let $p$ be a non-wandering point of a $C^r$ diffeomorphism $f$ of a compact manifold, with $r \ge 2$. Can $f$ be $C^r$-approximated by diffeomorphisms $g$ for which $p$ is a periodic point?
2. Problem 11: is one-dimensional dynamics generally hyperbolic? In particular, can a complex polynomial be approximated by polynomials of the same degree all of whose critical points tend to attracting periodic orbits?
3. Problem 12, centralisers of diffeomorphisms: can every $C^r$ diffeomorphism of a compact manifold be $C^r$-approximated by one that commutes only with its own powers?
4. Problem 13: Hilbert's sixteenth problem, on the number and position of the limit cycles of polynomial vector fields in the plane.
5. Problem 14: do the Lorenz equations have a strange attractor of the type of the geometric Lorenz model?

## Remarks

Problem 10 is the higher-smoothness version of Pugh's [[closing lemma|E1Z444F6TV]], which settles the case $r = 1$; it remains open in general.
For Problem 11, density of hyperbolicity among real polynomials was proved by Kozlovski, Shen and van Strien.
The real quadratic case had been settled before by Lyubich and by Graczyk and Świątek.
The complex case is the density of hyperbolicity conjecture and is open.
Problem 12 was solved in the $C^1$ topology by Bonatti, Crovisier and Wilkinson.
Problem 13 is open, even for quadratic vector fields.
Problem 14 was solved positively by Tucker with a computer-assisted proof; see [[Smale's fourteenth problem|VDVQD2HFSD]].

## Explanations

### L1 human

Smale chose a few questions that he thought would shape mathematics in the new century, following Hilbert's example of 1900.
Several of them ask how typical systems behave over long times.
Can recurrence be turned into periodicity by tiny smooth changes?
Is chaos rare or common in one-dimensional maps?
Does the famous Lorenz system really have the butterfly-shaped attractor seen on computers?
Some of these questions have been answered, and others are still open.