# Entropy

## Problem 154

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

## 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 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 143

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

## 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 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 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 133

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

## Problem 131

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

## 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 121

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

## 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 115

Are analytic maps $h$-expansive?

## 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 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 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 91

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

## 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 86

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

## Problem 78

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

## 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 76

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

## 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 69

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

## Problem 67

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

## 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 55

Entropy of automorphisms in algebra (groups, rings).

## 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 52

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

## 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 47

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

## Problem 42

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

## 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 40

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

## 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 37

Are geodesic flows $h$-expansive?

## Problem 34

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

## 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 28

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

## Problem 26

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

## Problem 23

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

## Problem 19

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

## 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 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 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?