10
votes

Accepted

### Ergodicity and mixing of geodesic and horocyclic flows

The algebraic approach is just one of many ways to deal with the geodesic and horocycle flows. Also your quick summary does not really pay tribute to the many ways to deal with it using "...

6
votes

Accepted

### How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?

The hyperbolic dimension of $f$ is 1 and its maximal hyperbolic set is the unit circle $\mathbb{S}^1$.
First we show that a hyperbolic set for $f$ must be contained in the unit circle $\mathbb{S}^1$.
...

5
votes

### How can I prove that the suspension of an Anosov diffeomorphism is an Anosov flow?

The short answer, as pointed out in the comments, is that yes, contraction at regular intervals is enough, because the definition of Anosov requires that there are $C>0$ and $\lambda\in (0,1)$ such ...

4
votes

### When entropy SRB measure is zero

The SRB measure is always isomorphic to a Bernoulli scheme (up to a period) and hence has positive entropy.
Regarding continuity properties of entropy, we have upper semicontinuity whenever the map ...

4
votes

### Can the Reeb foliation of $S^3$ be realized as stable manifold foliation of a smooth hyperbolic discrete dynamic on $S^3$?

No, because it has a unique leaf that is a torus. That can't be a stable or unstable manifold; one reason is that it can't get either bigger or smaller under the action of the hyperbolic dynamical ...

4
votes

Accepted

### Non-absolutely continuous foliation

I don’t have the Handbook near me at the moment so I can’t look at the example you mention, but at https://vaughnclimenhaga.wordpress.com/2013/11/24/fubini-foiled/ I describe a construction of Milnor’...

4
votes

### Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$

Here is an answer of sorts; it is not completely canonical though. First of all, you have to pick a conformal or hyperbolic structure on the fiber $\Sigma$. This can be made almost canonical, since ...

3
votes

### Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$

The pseudo-Anosov homeomorphism is a diffeomorphism away from finitely many singular points, so in general will not be a smooth diffeomorphism. However, for certain fibered knots there are not ...

3
votes

Accepted

### Invariant measure of geodesic flow on unit tangent bundle of a modular surface

$\newcommand{\diff}{\mathrm{d}}$This is a heavily edited version of my (a bit right, but mostly wrong) previous answer. I think that this is now correct (even got the signs right!).
$\newcommand{\HH}...

2
votes

Accepted

### Question on a proof of density of periodic orbits

If the direction of a geodesic through $p$ is far from the radial direction, it looks something like this. The only geodesics through $p$ that are long are close to the radial direction. Getting ...

2
votes

### Quantitative approximation of invariant measures by periodic ones

For a result of a somewhat similar type, check the paper "Rate of approximation of minimizing measures" by Bressaud and Quas, 2007, especially Theorem 4. In their case the dynamics is a subshift of ...

2
votes

### Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

I think before discussing about the regularity of holonomies of $W^s$, one needs to discuss the existence of the foliation $W^s$.
In the above setting there are algebraic examples where the ...

2
votes

Accepted

### Example of zero Lyapunov exponentes

I’m guessing you mean the base dynamical system to be a Bernoulli shift with the unperturbed cocycle being $A(x)=A_{x_0}$? For the second one, no perturbation is necessary. The Lyapunov exponents are ...

1
vote

Accepted

### Existence of center-stable manifold when the Jacobian is singular?

For both manifolds we do not need the strong invertibility. Moreover, for $W^{cs}$ this is stated in Exercise III.2, p.68 from the mentioned monograph. However, I will give below a more geometrical ...

1
vote

Accepted

### Examples of hyperbolic set and J-stable sets

Hyperbolic functions - for example, quadratic polynomials with an attractive periodic point - are examples of maps that are J-stable. The notion of J-stability arises from the famous article of Mañe, ...

1
vote

Accepted

### $C^1$ partially hyperbolic diffeomorphism have Hölder stable holonomies (reference request)

The preprint I looked for was "FLAVORS OF PARTIAL HYPERBOLICITY" by F. Abdenur and M. Viana.

1
vote

### The continuity of the the stable and unstable in definition of hyperbolic sets for flows

Yes, it follows from the growth conditions. It should be done in standard texts (e.g. Katok-Hasselblatt) but I have not checked. Indeed, it is a general fact about dominated splittings.
In fact, the ...

1
vote

### Continuity of Lyapunov spaces

This is a standard argument in Pesin Theory, which I'll sketch below leaving details to the reader. Let's pretend $M = \mathbb T^d, d \geq 2$, so that I don't have to bother with charts (the proof is ...

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