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This website is dedicated to the memory of Rufus Bowen (1947-1978). On it, you will find an interactive interface to the 157 mathematical problems he left behind in his notebook. The mission of this site is to continue the discussion of these problems by allowing users to enter comments. Each problem has been deciphered and transcribed below and is presented along with comments on its current status. To view a problem and previously entered comments, simply click on the problem number below.

In order to add a comment, we ask that you first create an account (using your own name as a userid in order to properly attribute comments). After logging in, you then click on “add a comment’’ for a given problem, enter your comment in a text box which will open and then click on the Save button. If you enter your comment in latex, it will automatically render when you click on Save. The site includes a Bibliography which will expand and grow as users contribute to it and and more comments are made.

Membership and comments on this site are moderated and we ask for your patience with this. This site is a work in progress and will change and improve as time goes on. If you have any questions or comments about the site, please check the Contact/Help page, and use the form on that page if you can't find the answer you are looking for.

Acknowledgements: Many thanks to Caroline Series for scanning in the notebook and to Francois Ledrappier for tex’ing up the problems and inputting the first set of comments.

The image of Rufus's notebook above is linked to a page of high resolution images of the individual pages can be viewed as a gallery or downloaded [pdf]

Some of the problems below are terse and can be difficult to interpret, because they were recorded by Rufus as notes to himself. In many cases the necessary context can be found in the comments section of each problem, but if you think you can help clarify a point or if you spot a mistake, please help by leaving a comment.

Problem 1

To what extent does the gradient flow near a critical point depend on the metric?

Problem 2

Topological Rokhlin's Theorem.

Problem 3

Geometric' proof that weak mixing implies mixing for a full set of $t$.

Problem 4

Classification of singularities by the local properties of the gradient flow.

Problem 5

Homogenous dynamics

1. Implications among
• unique ergodicity
• minimality
• entropy zero plus ergodicity
2. Simple or semi-simple case
• Which one-parameter subgroups are unstable/stable foliations for some ergodic affine?
• Try a).
3. Relate dynamical properties to representations of the group.
4. K-property implies Bernoull?
5. Weak mixing plus center s.s. implies Bernoulli? [For parts d and e, try nilmanifolds first]
6. Ergodic implies there is a unique measure of maximal entropy?

Problem 6

Zeta function for Axiom A flows and systems

1.  topological identification (try 1-dimensional $\Omega$ first); conjugacy invariance of $\zeta (0)$.
2. For $C^\infty$ flows, $\zeta (s)$ has a meromorphic extension to the complex plane.

3. Connection with Laplacian vs. geodesic results; automorphic forms.

4. Anosov actions.

Problem 7

Structure of basic sets

1. Classification via $(R,A)$
2. Local Axiom A implies embeddable

3. existence of canonical coordinates implies embeddable (compact abelian group actions ? are $\Omega$'s).

4. Phantom homology groups -shift equivalence of induced maps!

5. dim $\Omega$?; when is the quotient a manifold?

Problem 8

Non Axiom A examples. Newhouse, Abraham-Smale, Simon, Lorenz, billiards.

1. Axiomatic description
2. Statistical properties
3. For all $\epsilon$, there exists a horseshoe $X_\epsilon$ inside with $h(f|X_\epsilon) \geq h(f) - \epsilon$
4. Statistical properties of Lorenz in particular
5. Any specification type property

Problem 9

Unique ergodicity of $W^u$ for partially Anosov diffeos.

Problem 10

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

Problem 11

Nonalgebraic Anosov diffeos. Classify 3-dimensional Anosov flows; Is the variable curvature surface geodesic flow conjugate to constant curvature?

Problem 12

Shub's entropy conjecture: $h(f) \geq \log |\lambda|$

1.  for diffeos
2.  $\Omega$ finite plus hyperbolic
3.  Axiom A with cycles.

Problem 13

In Parry's Conjugate to linear' paper, what are the properties of the constructed measure? Does this work for equilibrium states too?

Problem 14

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

Problem 15

Renewal theorems for dependent random variables.

1.  Derive as a motivation for Axiom A flow mixingness
2.  How fast is the mixing for Axiom A flows?

Problem 16

Brownian motion or diffusion given a flow

Problem 17

Symbolic dynamics for billiards

Problem 18

Interpret $-\log \lambda^u$ as a potential function  (Kolmogorov's   idea on surfaces of negative curvature)?

Problem 19

Can you construct some Banach space so that $h_\mu$ is an eigenvalue of some canonical operator?

Problem 20

Continuous systems in statistical mechanics. Is there a topological dynamics formulation?

Problem 21

Assume $\varphi _t$ $C$-dense. If $\nu$ is $\varphi _1$ invariant is $\nu$ $\varphi$-invariant?

Problem 22

Canonical embedding of Axiom A $\Omega _i$

Problem 23

Canonical $C^0$ perturbation of Anosov diffeo to 0-dimensional $\Omega _i$'s with the same entropy

Problem 24

Bifurcation of Axiom A in terms of symbols.

Problem 25

This problem was crossed out.

Problem 26

Entropy in Hamiltonian case; for P.D.E.'s? Relation to O.D.E.'s?

Problem 27

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

Problem 28

Kupka-Smale plus $h(f) >0$ forces homoclinic points.

Problem 29

Find Axiom A infinite attractor in some O.D.E. on $\mathbb{R}^3$ (quadratic).

Problem 30

Fixing compact manifold $M$, what are the possible behaviors of the geodesic flows for all Riemannian metrics? For instance, if $\pi _1(M) = 0 ,$ does some geodesic flow have entropy 0?

Problem 31

Anosov diffeos

1. Hypothesis on $H_1(M)$
2. Fixed points
3. $\Omega = M$

Problem 32

Classify symbolic systems with specification.

Problem 33

Define $P(g)$, equilibrium states for certain noncontinuous $g$.

Problem 34

If $\mu$ is an equilibrium state for some continuous $g$ on $\Sigma _N^+$, is $h_\mu >0$?

Problem 35

Ergodic non hyperbolic automorphisms of $\mathbb{T}^n$. Are they quotients of subshifts of finite type? Do they satisfy specification?

Problem 36

If a geodesic flow is expansive, is it an Anosov flow?

Problem 37

Are geodesic flows $h$-expansive?

Problem 38

Let $\mathcal M$ be the space of Riemannian metrics on $M$ with volume 1.  What is $\{ h(\varphi_1), \varphi {\textrm { geodesic flow of }}\; g: g \in \mathcal M\}$? What is the relation with topological invariants of $M$?

Problem 39

Ambrose-Kakutani Theorem for $\mathbb{R}^n$ actions

Problem 40

Entropy of automorphisms of $C^\ast$-algebras

Problem 41

Is $\varphi_1$ a continuity point for the entropy as a function of diffeos when $\varphi_t$ is Axiom A flow? an Anosov flow?

Problem 42

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

Problem 43

Definition of Gibbs measures for homeomorphisms? Relation to equilibrium states?

Problem 44

How big is the set of equilibrium states of a function $g$ in the set of invariant measures?

Problem 45

Any local' invariant (near fixed points) which are entropy-like.

Problem 46

Equilibrium states for 1-dimensional quantum lattice systems without finite range

Problem 47

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

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 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 50

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

Problem 51

Look for invariant measures of some standard foliations.

Problem 52

Define $\Omega$(foliation). Does $h >0$ make sense?

Problem 53

This problem was crossed out.

Problem 54

$C$-dense (mixing) Axiom A flows

1. speed of mixing
2. asymptotic expression for the number of periodic orbits
3. is $\varphi_1$ intrinsically ergodic?
4. direct proof of mixing of measures
5. analogue of $h(f) \geq \log |\lambda|$
6. understand det$(Id - A)$ as an invariant; relation to $\zeta (0)$
7. stability of $C$-density for attractors
8. condition on $g$ so that $\Sigma_A (g)$ is analytically or $C^\infty$ embeddable as a basic set.
9. can a closed orbit of an Anosov flow be null homotopic?

Problem 55

Entropy of automorphisms in algebra (groups, rings).

Problem 56

Is the horocycle flow an expansive flow?

Problem 57

$\ell (f^n \gamma)$ grows slowly with $n$ for many curves $\gamma$ and Axiom A diffeos $f$.

Problem 58

Is Gutzwiller's example an Anosov flow?

Problem 59

Computer programs for Axiom A attractor.

Problem 60

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

Problem 61

Rokhlin theorem for countable pseudo group actions. Ergodic Theorems and averaging procedures.

Problem 62

Covering space for $\Sigma_A \to \mathbb{T}^2$ corresponding to $\mathbb{R}^2 \to \mathbb{T}^2$

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 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 65

Foliation ergodic theory

1. Ambrose Kakutani (in particular, question 39)
2. Does mixing make any sense? (use category, differentiability, $C^\infty$, analytic structure)
3. Averaging procedure difficulties:
• ergodic theorems, existence of invariant measures, ergodic decomposition, unique ergodicity and uniform convergence.
• polynomial growth  ...
4. Look at some specific foliations
5. Plante's stuff on connections with homology
6. Does pointwise entropy make sense?

Problem 66

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

Problem 67

Correspondence principle of quantum mechanics. Investigate for some simple mechanical systems. Is h-expansiveness related to quantum ....?

Problem 68

Electric circuits

• Analogue computer for finding Axiom A examples
• Is noise sometimes due to hyperbolicity in the dynamics?

Problem 69

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

Problem 70

Classify all Anosov systems or attractors (which $\Omega_i$ can occur as attractors?)

Problem 71

Cancellation of $\Omega_i$. Simplest $f$ in an iosotopy class.

Problem 72

This problem was crossed out.

Problem 73

If $f$ is Anosov and $g \sim f$, does $h(g) \geq h(f) ?$

Problem 74

If $f$ is Axiom A, is there an Axiom A $g$, $C^0$ near $f$ with dim$\Omega (g) = d$ and $h(f) = h(g) ?$

Problem 75

Conditions on $M$ to admit Anosov $f$

Problem 76

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

Problem 77

Conjugacy between topology and measure theory

a. Weakest notion such that h(f) is an invariant

b. Entropy-conjugacy + equivalence on Baire sets; what are the equivalence relations on homeomorphisms or maps on $S^1$ and subshifts?

Problem 78

Calculate $h$ for O.D.E. systems on $\mathbb{R}^n$, e.g. linear equations first

Problem 79

Infinite measure space automorphisms

Problem 80

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

Problem 81

Geometric proof of unique ergodicity for irrational rotation of $\mathbb S^1$

Problem 82

(Plante) A codimension one minimal foliation has at most one invariant measure

Problem 83

Unstable foliations of Anosov diffeos are given by some nilpotent group action.

Problem 84

Invariant or approximatively invariant finite dimensional subspaces for Navier-Stokes equation

Problem 85

codon frequencies via equilibrium states for `some potential''?

Problem 86

Algebraic varieties. Weil conjecture, cohomology. Any entropy here? Any relation to homology eigenvalues?

Problem 87

If a translation by a group element on $G/\Gamma$ is minimal, is that element nilpotent in $\mathfrak G ?$ (i.e. has 0 entropy)

Problem 88

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

Problem 89

For Anosov flow $\varphi_t$ on $M$, try to approximate curves in $M$ by pseudo-orbits and compute $\pi_1(M)$ . . . as in Morse theory.

Problem 90

If $f$ is Anosov on $M$ and $\tilde M$ contractible, what does $H^k(M)(\sim H^k (\pi _1(M)) )$ tell you via $f_\ast$ eigenvalue information? (See [1], pp. 200-202)

Problem 91

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

Problem 92

Among degree $n$ polynomial maps of $[0,1]$ to itself, are Axiom A open and dense. Do bad ones form a stratified set? ...

Problem 93

This problem isn't legible.

Problem 94

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

Problem 95

Markov partitions for algebraic geometry examples (Ruelle).

Problem 96

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

Problem 97

Example of a non-ergodic $C^1$ Anosov diffeo on $\mathbb{T}^2$ preserving Lebesgue measure.

Problem 98

This problem is not clearly legible, but it is thought to say

Unique ergodicity in Lie groups. Case with finite area instead of compact homogeneous spaces

Problem 99

(Doug (Lind?)) Find open partitions in $\Sigma _{1/2,1/2}$ that are not weakly Bernoulli. Find invariants of finitary codes.

Problem 100

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

Problem 101

If a $C^1$ Anosov preserves a smooth measure, is it an equilibrium state for $- \log \lambda ^u ?$

Problem 102

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

Problem 103

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

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 105

This problem isn't clearly legible, but it is thought to say: Sullivan's stacks of coins problem

Problem 106

Is the multiplicity of $1$ as an eigenvalue of $A$ a flow conjugacy invariant of $\Sigma _A$? How about $\Pi_{\lambda_i \not = 1} (1- \lambda_i)$?

Problem 107

Embed automorphisms of compact groups as basic sets.

Problem 108

Let a group $G$ be given by a generator $S$ and relations. Consider the set $V$ of reduced  (one-sided or two-sided) infinite words. What is $V$?  Is it intrinsically ergodic? What is the entropy?

Problem 109

(Thom) Look at Markov partitions on $\mathbb{T}^n$ when all $\lambda_i$ are  distinct and real.

Problem 110

Topological entropy of the Frobenius map of an algebraic variety $V$. Related to Dim $V$, to the log of the radius of convergence of the zeta function? Relations to zeta functions and to Weil conjectures.

Problem 111

Central Limit Theorem for $\beta$-transform $x \mapsto (\beta x)$.

Problem 112

For a hyperbolic attractor $\Lambda$ of dimension $r$, does $W^s(x) \cap \Lambda$contain a disk of dimension $k := r- Dim W^u (x)?$

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 114

$det (I - A)$ as a group invariant for the weak foliation $W^{wu}$, seen as a subgroup lying in some $H_m (M, \mathbb S^1)$?

Problem 115

Are analytic maps $h$-expansive?

Problem 116

For a subshift $\Sigma _A$, $A$ on stable torus or stable $\mathbb{R}^n$ is an invariant. Describe it invariantly.

($\mathbb{R}^n = ?$, $A = ?$)

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 118

This problem was crossed out.

Problem 119

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

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 121

Calculate $h_\mu$ for $G/\Gamma$ finite measure, non compact.

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 123

Cancellation theorem for basic sets. Analogue of cobordism theorem.

Problem 124

Weil conjecture for basic sets.

Problem 125

Embedding algebraic variety over $Z_p$ into  a basic set.

Problem 126

Put orientation into the $\zeta$ function of flows. What should $\zeta (0)$ be? Does $\zeta (0)$ depend only on $H_\ast (M_0, M_{sing}) ?$.

Problem 127

This problem was crossed out.

Problem 128

When are suspensions of $R_\alpha$ and $R_\beta$ under bounded functions isomorphic?

Problem 129

See the entropy of the geodesic flow as the rate of growth of eigenfunctions for some operator in momentum space. Related to Fourier transform of Laplacian on the manifold?

Problem 130

Which surfaces and which homotopy classes of homeos admit expansive homeos? distal homeos?

Problem 131

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

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 133

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

Problem 134

1. If $\varphi _t$ is flow on a homogeneous space $G/ \Gamma$ with positive entropy, then there exists a compact $\varphi _t$ invariant section for the action of $N$
2. If the flow has entropy 0 and is ergodic, does this mean that there is  no $N$?

Problem 135

Horocycle and geodesic flows for $SL(2,R)/SL(2,Z)$:

-- min u.e. almost

-- something about symbolic dynamics and continued fractions?

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 137

Ergodic smooth representative of a Dehn twist. With smooth invariant measure of positive entropy. Of maximal entropy.

Problem 138

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

Problem 139

Are polynomial growth foliations hyperfinite? Is $W^{ws}$ on $\Sigma ^+_{\{0,1\}}$ Borel hyperfinite?

Problem 140

Let $g : G/\Gamma \to G/\Gamma$ come from an automorphism $g$ of $G$. Is the entropy minimal in its homotopy class?  Is entropy a complete invariant for automorphisms of infranilmanifolds?

Problem 141

Same setting [as question 140]. Is the $GR$ (of $g$ on $\Gamma$) an algebraic number?

Problem 142

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

Problem 143

Is $GR (\alpha)$ an algebraic integer for an automorphism $\alpha$ of solvable group?

Problem 144

Question 136 with expansive instead of hyperbolic.

Problem 145

This problem was crossed out.

Problem 146

1. Subshifts of finite type have good quotients with fixed points.
2. Given a periodic point $p$  in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
3. If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
4. Do two subshifts of finite type with the same entropy have a common good quotient?
5. $\Sigma _A, \Sigma _B$ aperiodic, $h (\sigma _A ) < h(\sigma_B).$ Does $\sigma_A |\Sigma_A$ embed in $\sigma_B |\Sigma_B$?

Problem 147

(Guillemin-Kazhdan) Invariant distributions for geodesic flows. Are they approximated by periodic orbit measures?

Problem 148

$\Gamma$ a Kleinian group with limit set $\Lambda$. Specification when there are no parabolic nor elliptic elements.

Problem 149

(Handel) If there exists a cross section for for all minimal sets of a flow, then there exist a global cross-section.

Problem 150

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

Problem 151

On $\Sigma _A^+$, define $x \sim y$ if $\sigma ^n x = \sigma^m y$ for some $n,m$. Find topological invariants for $(\Sigma _A^+, \sim )$.

Problem 152

$T:M \to M \; C^\infty$ Anosov. $f \in C^r$ and $f(x) = u(x) - u(Tx) .$   Does  $u \in C^r?$ ($r\geq 2$).

Problem 153

$\varphi _t: T^1M \to T^1M$ Anosov geodesic flow and $V: M \to \mathbb{R}$ such that $\int V(\pi \varphi_t x ) = 0$ on every closed geodesic. Is $V$ identically $0$?

Problem 154

The geodesic flow on surfaces of higher genus always has positive measure-theoretic entropy.

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 156

For a Kleinian group $\Gamma$, is the Hausdorff dimension of $\Lambda (\Gamma) <2$ if $\Lambda(\Gamma)$ is not the whole sphere?

(Note, this problem was added by the editor to the end of Rufus' last paper [1])

References

1. [bowen1979hausdorff] Bowen R.  1979.  Publications Mathématiques de l'IHÉS. 50:11–25.

Problem 157

On the closure $\overline T$ of Teichmüller space, consider a continuous parametrization $\overline T \times \Sigma_A^+ \to \mathbb S^2$ such that Image ($t, \Sigma_A^+ = \Lambda(\Gamma_t)$). Is the Hausdorff dimension of $\Lambda(\Gamma_t)$ continuous in $t \in \overline T$?

(Note, this problem was added by the editor to the end of Rufus' last paper [1])

References

1. [bowen1979hausdorff] Bowen R.  1979.  Publications Mathématiques de l'IHÉS. 50:11–25.

Problem 158

Entropy = 0

1. Ergodic group rotation is l.b. (loosely bernoulli)
2. Smooth system finitely many fixed points are l.b. (Katok)
3. distal?