# SmoothDyn

## Problem 155

(Thurston-Sullivan?) Are all smooth actions of $\Phi _g$ on $\mathbb S^1$ which are topologically conjugate to a standard one differentiably conjugate to the standard one?

## Problem 150

Is there an expansive homeo of $\mathbb S^2$?

## Problem 142

Can the  adding machine be an invariant set for a $C^2$ mapping of the 2-disk? of the $n$-disk?

## Problem 138

Billiards in right angle triangles. Find examples when it is ergodic.

## Problem 136

For most $C^2$ maps $f : [0,1] \to [0,1]$, for all $\epsilon >0$ there is a hyperbolic set $\Lambda$ such that $h(f|\Lambda) \geq h(f) - \epsilon.$ (See question 8c.)

## Problem 133

$C^ \infty$ diffeo of the 2-disk preserving a smooth measure $\mu$ with $h_\mu >0$? An ergodic example?

## Problem 132

$\dot X = Q(X)$ on $\mathbb{R}^3$, where $Q$ is quadratic. Is there a condition on the coefficients which guarantees a homoclinic point (a complicated attractor)? E.g. like for Reynolds numbers?

## Problem 131

$h(f)$ given by periodic points for generic $C^1$ map from $I$ to $I$? (or generic continuous map?)

## Problem 122

Fibration Theorem for $\log |\lambda|$? If $M = \cup _\alpha N_\alpha, N_\alpha$ submanifolds, with $f(N_\alpha ) = N_\alpha$, is the spectral radius of $f_\ast$ on $M$ $\le$ the sup of the spectral radius of $f_\ast$ on $N_\alpha$'s?

If $f|N_\alpha$ is an isometry, does the spectral radius of $f_\ast$ on $M$ $= 1$? Are all distal diffeos built up this way, i.e. by extensions where homology works?

Note: There is another Problem 122 that was crossed out.

## Problem 120

Entropy 0 for differentiable action of $G = \mathbb{R}^n$ or $\mathbb{Z}^n (n \geq 2).$ For nilpotent $G$. General Lie group $G$?

## Problem 119

(Thom) $Grad F$ for real analytic $F:\mathbb{R}^n \to \mathbb{R}$. Stratification of orbits near a singular point.

## Problem 117

(Sullivan) Show that $|\lambda| = 1$ for $f: M \to M$, where $f$ is  $C^1$ and distal and $\lambda$ is an eigenvalue of $f_\ast$. Is $f \sim g$ for some Morse Smale $g$?

## Problem 115

Are analytic maps $h$-expansive?

## Problem 113

For a rational function $f(z)$ giving a degree $n$ map $z \mapsto f(z)$ of $\mathbb S^2$, does this map have entropy $\log n?$

($\ge$ by Misurewicz and = a.e. by Guckenheimer)

## Problem 104

Note: $\frac {1}{1-t} \; = \; \Pi _{n= 0}^\infty (1 +t^{2^n})$ in $Z[[ t]]$. Is there a $C^2$ map $\mathbb D^2 \to \mathbb D^2$ so that.....

## Problem 103

How can you write $1 +t +t^2 \; = \; \Pi _{i=0 }^\infty (1\pm t^{n_i})$ in $Z[[ t]] ?$

## Problem 102

Is the false zeta function of a basic set ....? Define false zeta function for a flow basic set

## Problem 100

This problem isn't clearly legible, but it is thought to say: Symbolic dynamics for Abraham-Smale, Newhouse, Lorentz

## Problem 96

KAM Theorem using $\mu$ uniquely ergodic  flows on $M$ without assuming $M = T^n$

## Problem 94

Does every manifold $M^n, n\geq 3$ admit a smooth Bernoulli flow (Ruelle)?

## Problem 91

Is the topological entropy continuous on $C^1$ expanding maps of the interval? (a.e. continuous?)

## Problem 88

Index 0 fixed point of diffeo is removable by small perturbation (Hirsch)

## Problem 80

Reddy examples of expansive maps. Related to Anosov diffeos. Are expansive diffeos likely to be Anosov?

## Problem 76

Does $h(f)$ have a minimum in isotopy class?

## Problem 69

Is $h: Diff ^r \to \mathbb{R}$ generically continuous for some $r$?

## Problem 66

Central Limit Theorem, other strong statistics near an attractor of a diffeo.

## Problem 64

Entropy of group actions. There is no smooth $\mathbb{R}^n$- (or $\mathbb{Z}^n$-) action with positive entropy when $n >1$. Is this true for all Lie groups (or lattices) of dimension greater than 1? Try $N$ nilpotent.

## Problem 63

$C^r$ diffeos that are not $C^{r+1}$ qualitatively. Find a $C^1$ diffeo $f : V \to V$ and a $C^1$ embedding $g:V \to M$ and $\tilde f$ extending $f$ to $M$ such that $\tilde f$ is $C^2$ on $M$, but $\tilde f |V$ is qualitatively not $C^2$ (i.e. the qualitative behavior of $\tilde f|V$ is due to the irregularity of $V$ but not of $\tilde f$)

## Problem 60

Study flows $H = V(r) + \frac{1}{2} mv^2$ for various (?) continuous $V(r)$. Statistical mechanics literature (Hénon -?, Toda,...)

## Problem 50

Is there a transitive/ergodic diffeomophism on $\mathbb S^2, \mathbb D^2$?

## Problem 49

Does minimal or uniquely ergodic for a diffeo  $f$ implies $h(f) = 0$ (try homeo case too)?

1. Is there a minimal diffeo homotopic to $\left( \begin{matrix} 2 & 1\\ 1& 1 \end{matrix} \right) \times Id.$ on $\mathbb T^3$?
2. (Seifert conjecture) Minimal flow on $\mathbb S^3$.

## Problem 48

Suppose $F: C \to \mathbb{R}, C$ the Cantor set, has bounded total variation. Is there a homeo $g : [0,1] \to [0,1]$ and a diffeo (Lipschitz, maybe) $f:[0,1] \to \mathbb{R}$ such that $F = f\circ g |C.$

## Problem 47

Any entropy-like invariant for singularity of diff. maps?

## Problem 42

Is $h(\varphi_t | E)$ differentiable in $E$ for Hamiltonian flows? Any relation to classical or quantum statistical mechanics?

## Problem 27

Construction of a 2-dimensional Hamiltonian diffeo. with an ergodic set of positive measure.

## Problem 16

Brownian motion or diffusion given a flow

## Problem 14

Suspensions of diffeos. -  Are they generically not  (conjugated to) constant time suspensions? what is the strongest statement for Axiom A attractors?

## Problem 10

Statistics plus dynamics of transformations of $[0,1]$ - 'non-linear' $\beta$-expansions like examples.