# 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?