To what extent does the gradient flow near a critical point depend on the metric?
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This website is dedicated to the memory of Rufus Bowen (19471978). 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.
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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.
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.
`Geometric' proof that weak mixing implies mixing for a full set of $t$.
Classification of singularities by the local properties of the gradient flow.
Homogenous dynamics
 Implications among
 unique ergodicity
 minimality
 entropy zero plus ergodicity
 Simple or semisimple case
 Which oneparameter subgroups are unstable/stable foliations for some ergodic affine?
 Try a).
 Relate dynamical properties to representations of the group.
 Kproperty implies Bernoull?
 Weak mixing plus center s.s. implies Bernoulli? [For parts d and e, try nilmanifolds first]
 Ergodic implies there is a unique measure of maximal entropy?
Zeta function for Axiom A flows and systems
 topological identification (try 1dimensional $\Omega $ first); conjugacy invariance of $\zeta (0)$.

For $C^\infty $ flows, $\zeta (s) $ has a meromorphic extension to the complex plane.

Connection with Laplacian vs. geodesic results; automorphic forms.

Anosov actions.
Structure of basic sets
 Classification via $(R,A)$

Local Axiom A implies embeddable

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

Phantom homology groups shift equivalence of induced maps!

dim $\Omega$?; when is the quotient a manifold?
Non Axiom A examples. Newhouse, AbrahamSmale, Simon, Lorenz, billiards.
 Axiomatic description
 Statistical properties
 For all $\epsilon$, there exists a horseshoe $X_\epsilon$ inside with $ h(fX_\epsilon) \geq h(f)  \epsilon$
 Statistical properties of Lorenz in particular
 Any specification type property
Unique ergodicity of $W^u$ for partially Anosov diffeos.
Statistics plus dynamics of transformations of $[0,1]$  'nonlinear' $\beta$expansions like examples.
Nonalgebraic Anosov diffeos. Classify 3dimensional Anosov flows; Is the variable curvature surface geodesic flow conjugate to constant curvature?
Shub's entropy conjecture: $h(f) \geq \log \lambda $
 for diffeos
 $\Omega $ finite plus hyperbolic
 Axiom A with cycles.
In Parry's `Conjugate to linear' paper, what are the properties of the constructed measure? Does this work for equilibrium states too?
Suspensions of diffeos.  Are they generically not (conjugated to) constant time suspensions? what is the strongest statement for Axiom A attractors?
Renewal theorems for dependent random variables.
 Derive as a motivation for Axiom A flow mixingness
 How fast is the mixing for Axiom A flows?
Interpret $\log \lambda^u$ as a potential function (Kolmogorov's idea on surfaces of negative curvature)?
Can you construct some Banach space so that $h_\mu $ is an eigenvalue of some canonical operator?
Continuous systems in statistical mechanics. Is there a topological dynamics formulation?
Assume $\varphi _t $ $C$dense. If $\nu $ is $\varphi _1 $ invariant is $\nu $ $\varphi $invariant?
Canonical embedding of Axiom A $\Omega _i$
Canonical $C^0$ perturbation of Anosov diffeo to 0dimensional $\Omega _i$'s with the same entropy
Entropy in Hamiltonian case; for P.D.E.'s? Relation to O.D.E.'s?
Construction of a 2dimensional Hamiltonian diffeo. with an ergodic set of positive measure.
KupkaSmale plus $h(f) >0 $ forces homoclinic points.
Find Axiom A infinite attractor in some O.D.E. on $\mathbb{R}^3$ (quadratic).
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?
Anosov diffeos
 Hypothesis on $H_1(M)$
 Fixed points
 $\Omega = M$
Classify symbolic systems with specification.
Define $P(g)$, equilibrium states for certain noncontinuous $g$.
If $\mu $ is an equilibrium state for some continuous $g$ on $\Sigma _N^+$, is $h_\mu >0$?
Ergodic non hyperbolic automorphisms of $\mathbb{T}^n$. Are they quotients of subshifts of finite type? Do they satisfy specification?
If a geodesic flow is expansive, is it an Anosov flow?
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 $M$?
AmbroseKakutani Theorem for $\mathbb{R}^n$ actions
Entropy of automorphisms of $C^\ast $algebras
Is $\varphi_1$ a continuity point for the entropy as a function of diffeos when $\varphi_t $ is Axiom A flow? an Anosov flow?
Is $h(\varphi_t  E)$ differentiable in $E$ for Hamiltonian flows? Any relation to classical or quantum statistical mechanics?
Definition of Gibbs measures for homeomorphisms? Relation to equilibrium states?
How big is the set of equilibrium states of a function $g$ in the set of invariant measures?
Any `local' invariant (near fixed points) which are entropylike.
Equilibrium states for 1dimensional quantum lattice systems without finite range
Any entropylike invariant for singularity of diff. maps?
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 \[ F = f\circ g C.\]
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 $\mathbb S^3$.
Is there a transitive/ergodic diffeomophism on $\mathbb S^2, \mathbb D^2$?
Look for invariant measures of some standard foliations.
Define $\Omega $(foliation). Does $h >0 $ make sense?
$C$dense (mixing) Axiom A flows
 speed of mixing
 asymptotic expression for the number of periodic orbits
 is $\varphi_1$ intrinsically ergodic?
 direct proof of mixing of measures
 analogue of $h(f) \geq \log \lambda $
 understand det$(Id  A) $ as an invariant; relation to $\zeta (0)$
 stability of $C$density for attractors
 condition on $g$ so that $\Sigma_A (g) $ is analytically or $C^\infty $ embeddable as a basic set.
 can a closed orbit of an Anosov flow be null homotopic?
$\ell (f^n \gamma) $ grows slowly with $n$ for many curves $\gamma$ and Axiom A diffeos $f$.
Study flows $H = V(r) + \frac{1}{2} mv^2$ for various (?) continuous $V(r)$. Statistical mechanics literature (Hénon ?, Toda,...)
Rokhlin theorem for countable pseudo group actions. Ergodic Theorems and averaging procedures.
Covering space for $\Sigma_A \to \mathbb{T}^2$ corresponding to $\mathbb{R}^2 \to \mathbb{T}^2$
$C^r$ 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 fV $ is due to the irregularity of $V$ but not of $\tilde f$)
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.
Foliation ergodic theory
 Ambrose Kakutani (in particular, question 39)
 Does mixing make any sense? (use category, differentiability, $C^\infty$, analytic structure)
 Averaging procedure difficulties:
 ergodic theorems, existence of invariant measures, ergodic decomposition, unique ergodicity and uniform convergence.
 polynomial growth ...
 Look at some specific foliations
 Plante's stuff on connections with homology
 Does pointwise entropy make sense?
Central Limit Theorem, other strong statistics near an attractor of a diffeo.
Correspondence principle of quantum mechanics. Investigate for some simple mechanical systems. Is hexpansiveness related to quantum ....?
Electric circuits
 Analogue computer for finding Axiom A examples
 Is noise sometimes due to hyperbolicity in the dynamics?
Is $h: Diff ^r \to \mathbb{R}$ generically continuous for some $r$?
Classify all Anosov systems or attractors (which $\Omega_i$ can occur as attractors?)
Cancellation of $\Omega_i$. Simplest $f$ in an iosotopy class.
If $f$ is Anosov and $g \sim f$, does $h(g) \geq h(f) ?$
If $f$ is Axiom A, is there an Axiom A $g$, $C^0$ near $f$ with dim$\Omega (g) = d$ and $h(f) = h(g) ?$
Conjugacy between topology and measure theory
a. Weakest notion such that h(f) is an invariant
b. Entropyconjugacy + equivalence on Baire sets; what are the equivalence relations on homeomorphisms or maps on $S^1$ and subshifts?
Calculate $h$ for O.D.E. systems on $\mathbb{R}^n$, e.g. linear equations first
Reddy examples of expansive maps. Related to Anosov diffeos. Are expansive diffeos likely to be Anosov?
Geometric proof of unique ergodicity for irrational rotation of $\mathbb S^1$
(Plante) A codimension one minimal foliation has at most one invariant measure
Unstable foliations of Anosov diffeos are given by some nilpotent group action.
Invariant or approximatively invariant finite dimensional subspaces for NavierStokes equation
codon frequencies via equilibrium states for ``some potential''?
Algebraic varieties. Weil conjecture, cohomology. Any entropy here? Any relation to homology eigenvalues?
If a translation by a group element on $G/\Gamma$ is minimal, is that element nilpotent in $\mathfrak G ?$ (i.e. has 0 entropy)
Index 0 fixed point of diffeo is removable by small perturbation (Hirsch)
For Anosov flow $\varphi_t $ on $M$, try to approximate curves in $M$ by pseudoorbits and compute $\pi_1(M)$ . . . as in Morse theory.
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. 200202)
References
Is the topological entropy continuous on $C^1$ expanding maps of the interval? (a.e. continuous?)
Among degree $n$ polynomial maps of $[0,1]$ to itself, are Axiom A open and dense. Do bad ones form a stratified set? ...
Does every manifold $M^n, n\geq 3$ admit a smooth Bernoulli flow (Ruelle)?
Markov partitions for algebraic geometry examples (Ruelle).
KAM Theorem using $\mu $ ergodic (??) flows on $M$ without assuming $M = T^n$
Example of a nonergodic $C^1$ Anosov diffeo on $\mathbb{T}^2$ preserving Lebesgue measure.
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
(Doug (Lind?)) Find open partitions in $\Sigma _{1/2,1/2}$ that are not weakly Bernoulli. Find invariants of finitary codes.
This problem isn't clearly legible, but it is thought to say: Symbolic dynamics for AbrahamSmale, Newhouse, Lorentz
If a $C^1$ Anosov preserves a smooth measure, is it an equilibrium state for $  \log \lambda ^u ?$
Is the false zeta function of a basic set ....? Define false zeta function for a flow basic set
How can you write \[ 1 +t +t^2 \; = \; \Pi _{i=0 }^\infty (1\pm t^{n_i}) \] in $Z[[ t]] ?$
Note: \[ \frac {1}{1t} \; = \; \Pi _{n= 0}^\infty (1 +t^{2^n}) \] in $Z[[ t]]$. Is there a $C^2$ map $\mathbb D^2 \to \mathbb D^2$ so that.....
This problem isn't clearly legible, but it is thought to say: Sullivan's stacks of coins problem
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) $?
Embed automorphisms of compact groups as basic sets.
Let a group $G$ be given by a generator $S$ and relations. Consider the set $V$ of reduced (onesided or twosided) infinite words. What is $V$? Is it intrinsically ergodic? What is the entropy?
(Thom) Look at Markov partitions on $\mathbb{T}^n$ when all $\lambda_i $ are distinct and real.
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.
Central Limit Theorem for $\beta$transform $x \mapsto (\beta x)$.
For a hyperbolic attractor $\Lambda$ of dimension $r$, does $W^s(x) \cap \Lambda $contain a disk of dimension $k := r Dim W^u (x)?$
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)
$det (I  A) $ as a group invariant for the weak foliation $W^{wu}$, seen as a subgroup lying in some $H_m (M, \mathbb S^1)$?
For a subshift $\Sigma _A$, $A$ on stable torus or stable $\mathbb{R}^n$ is an invariant. Describe it invariantly.
($\mathbb{R}^n = ?$, $A = ?$)
(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 $g$?
(Thom) $Grad F$ for real analytic $F:\mathbb{R}^n \to \mathbb{R}$. Stratification of orbits near a singular point.
Entropy 0 for differentiable action of $G = \mathbb{R}^n $ or $\mathbb{Z}^n (n \geq 2).$ For nilpotent $G$. General Lie group $G$?
Calculate $h_\mu $ for $G/\Gamma $ finite measure, non compact.
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 $fN_\alpha$ 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.
Cancellation theorem for basic sets. Analogue of cobordism theorem.
Embedding algebraic variety over $Z_p$ into a basic set.
Put orientation into the $\zeta$ function of flows. What should $\zeta (0)$ be? Does $\zeta (0)$ depend only on $H_\ast (M_0, M_{sing}) ?$.
When are suspensions of $R_\alpha $ and $R_\beta$ under bounded functions isomorphic?
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?
Which surfaces and which homotopy classes of homeos admit expansive homeos? distal homeos?
$h(f) $ given by periodic points for generic $C^1$ map from $I$ to $I$? (or generic continuous map?)
$\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?
$C^ \infty $ diffeo of the 2disk preserving a smooth measure $\mu $ with $h_\mu >0$? An ergodic example?
 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 $N$
 If the flow has entropy 0 and is ergodic, does this mean that there is no $N$?
Horocycle and geodesic flows for $SL(2,R)/SL(2,Z)$:
 min u.e. almost
 something about symbolic dynamics and continued fractions?
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.)
Ergodic smooth representative of a Dehn twist. With smooth invariant measure of positive entropy. Of maximal entropy.
Billiards in right angle triangles. Find examples when it is ergodic.
Are polynomial growth foliations hyperfinite? Is $W^{ws} $ on $\Sigma ^+_{\{0,1\}} $ Borel hyperfinite?
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?
Same setting [as question 140]. Is the $GR$ (of $g$ on $\Gamma$) an algebraic number?
Can the adding machine be an invariant set for a $C^2$ mapping of the 2disk? of the $n$disk?
Is $GR (\alpha) $ an algebraic integer for an automorphism $\alpha$ of solvable group?
Question 136 with expansive instead of hyperbolic.
 Subshifts of finite type have good quotients with fixed points.
 Given a periodic point $p$ in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
 If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
 Do two subshifts of finite type with the same entropy have a common good quotient?
 $\Sigma _A, \Sigma _B$ aperiodic, $h (\sigma _A ) < h(\sigma_B). $ Does $\sigma_A \Sigma_A$ embed in $\sigma_B \Sigma_B$?
(GuilleminKazhdan) Invariant distributions for geodesic flows. Are they approximated by periodic orbit measures?
$\Gamma$ a Kleinian group with limit set $\Lambda$. Specification when there are no parabolic nor elliptic elements.
(Handel) If there exists a cross section for for all minimal sets of a flow, then there exist a global crosssection.
Is there an expansive homeo of $\mathbb S^2$?
On $\Sigma _A^+$, define $x \sim y $ if $\sigma ^n x = \sigma^m y $ for some $n,m $. Find topological invariants for $(\Sigma _A^+, \sim )$.
$T:M \to M \; C^\infty $ Anosov. $f \in C^r $ and $f(x) = u(x)  u(Tx) .$ Does $u \in C^r?$ ($r\geq 2$).
$\varphi _t: T^1M \to T^1M$ 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 $0$?
The geodesic flow on surfaces of higher genus always has positive measuretheoretic entropy.
(ThurstonSullivan?) 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?
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
 [bowen1979hausdorff] Hausdorff dimension of quasicircles, , Publications Mathématiques de l'IHÉS, Volume 50, p.11–25, (1979)
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
 [bowen1979hausdorff] Hausdorff dimension of quasicircles, , Publications Mathématiques de l'IHÉS, Volume 50, p.11–25, (1979)
Entropy = 0
 Ergodic group rotation is l.b. (loosely bernoulli)
 Smooth system finitely many fixed points are l.b. (Katok)
 distal?