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 $\mathbb S^3$.


For b: [1], [2].


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 the first question, there are uniquely ergodic homeomorphisms of T^2 with positive entropy ( and minimal analytic diffeomorphisms with positive entropy in dimension 4 ( 

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. 

Add a new comment

Log in or register to post comments