(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?

# SmoothDyn

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

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

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

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.)

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

$\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?

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

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.

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

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

(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 $g$?

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)

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.....

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

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

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

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

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

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

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

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

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

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

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

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.

$C^r$ 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 $\tilde f$)

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

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

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

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

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 \[ F = f\circ g |C.\]

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

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

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

Brownian motion or diffusion given a flow

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

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