# Problem 49

## Primary tabs

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 $\mathbb S^3$.

## Tags

For b: [1], [2].

### References

1. [kuperberg1994smooth] Kuperberg K.  1994.  Annals of Mathematics. 140:723–732.
2. [kuperberg1996volume] Kuperberg G.  1996.  Commentarii Mathematici Helvetici. 71:70–97.

The Seifert conjecture asked if every non-singular flow on S3 has a periodic orbit. The answer is no by K. Kuperberg's construction.

The existence of a minimal flow on S3 is Gottschalk's conjecture and it is still an open problem.

### For a), unless I am missing

For the first question, there are uniquely ergodic homeomorphisms of T^2 with positive entropy (https://arxiv.org/PS_cache/math/pdf/0605/0605438v1.pdf) and minimal analytic diffeomorphisms with positive entropy in dimension 4 (https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-sys...).

For a), Franks' semiconjugacy gives a negative answer:

Take $F$ homotopic to $(2111)\times id$ and a lift of $\hat F$ to $R^2 x R.$ If $\{(x_n,t_n)\}$ is an $\hat F$ orbit, then $\{x_n\}$, its projection to $R^2$  is a bounded pseudo-orbit for the linear map $(2111)$ acting on $R^2$. Therefore, there is a unique orbit $\{y_n\}$ of $(2111)$ that shadows this pseudo-orbit. The map $x_0 \mapsto y_0$ is continuous, equivariant and bounded distance from the projection (therefore surjective). This is Franks' semiconjugacy which descends to a map $h: T^2 \times S^1 \to T^2$.

Now, the preimage of a fixed (or periodic) orbit of $(2111)$ in $T^2$ will give a closed invariant subset contradicting minimality.