Hyperbolic

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 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 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 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 144

Question 136 with expansive instead of hyperbolic.

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 125

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

Problem 124

Weil conjecture for basic sets.

Problem 123

Cancellation theorem for basic sets. Analogue of cobordism theorem.

Problem 119

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

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 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 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 111

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

Problem 107

Embed automorphisms of compact groups as basic sets.

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 103

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

Problem 102

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

Problem 101

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

Problem 99

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

Problem 97

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

Problem 94

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

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 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 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 83

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

Problem 80

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

Problem 75

Conditions on $M$ to admit Anosov $f$

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 73

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

Problem 71

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

Problem 70

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

Problem 68

Electric circuits

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

Problem 59

Computer programs for Axiom A attractor.

Problem 58

Is Gutzwiller's example an Anosov flow?

Problem 57

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

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 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 36

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

Problem 31

Anosov diffeos

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

Problem 29

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

Problem 28

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

Problem 24

Bifurcation of Axiom A in terms of symbols.

Problem 23

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

Problem 22

Canonical embedding of Axiom A $\Omega _i$

Problem 21

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

Problem 18

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

Problem 17

Symbolic dynamics for billiards

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 14

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

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 11

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

Problem 9

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

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 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 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 4

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

Problem 1

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