11
votes
Accepted
Applications of the Central Limit Theorem in dynamical systems
These limit theorems can be useful when studying systems preserving an infinite invariant measure.
For example, Jean Pierre Conze has used it in his paper "Sur un critere de recurrence en dimension ...
- 326
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 "...
- 18.5k
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 $\mathbb{S}^1$.
...
- 645
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 ...
- 8,782
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 ...
- 5,536
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 ...
- 9,749
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 ...
- 8,782
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’...
- 8,782
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}...
- 26.3k
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 ...
- 66.8k
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 ...
- 2,411
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 ...
- 22.5k
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 ...
- 940
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 ...
- 22.5k
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 ...
- 758
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, ...
- 6,455
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.
- 14.2k
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 ...
- 3,878
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 ...
- 475
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