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

# Entropy

$\varphi _t: T^1M \to T^1M$ 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 $0$?

- 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 $\sigma_B |\Sigma_B$?

Question 136 with expansive instead of hyperbolic.

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

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?

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.)

- 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 $N$?

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

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

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?

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

Entropy 0 for differentiable action of $G = \mathbb{R}^n $ or $\mathbb{Z}^n (n \geq 2).$ For nilpotent $G$. General Lie group $G$?

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)

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.

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?

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

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

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

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

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?

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

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

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

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

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

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.

Entropy of automorphisms in algebra (groups, rings).

$C$-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?

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

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 $\mathbb S^3$.

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

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

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

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

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 $M$?

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

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?

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

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

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

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

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

- for diffeos
- $\Omega $ finite plus hyperbolic
- Axiom A with cycles.

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

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

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

Homogenous dynamics

- Implications among
- unique ergodicity
- minimality
- entropy zero plus ergodicity

- Simple or semi-simple case
- Which one-parameter subgroups are unstable/stable foliations for some ergodic affine?
- Try a).

- Relate dynamical properties to representations of the group.
- K-property 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?