Questions tagged [anosov-systems]
The anosov-systems tag has no usage guidance.
19
questions
2
votes
0
answers
24
views
Regularity and rigidity of stable/unstable distribution for geodesic flow on noncompact negatively curved manifolds
For a volume-preserving $C^\infty$ Anosov flow on a three-dimensional compact Riemannian manifold, it was shown by Hurder & Katok that the Anosov foliations are always of class $C^{1, \alpha}$. ...
2
votes
1
answer
260
views
Anosov flow on the 2-sphere
Is there a simple proof that there is no Anosov flow on $S^2$? Where can I find it?
4
votes
1
answer
489
views
Question about an early result on the mixing of geodesic flows
Let $T_t$ be the geodesic flow on a surface $S$ of constant negative curvature, and let $M(f,t) := \langle \bar f \cdot (f \circ T_t) \rangle$, where $\langle f \rangle := \int_S f(x) d\mu(x)$ and ...
5
votes
1
answer
272
views
Smoothening pseudo-Anosov flows
A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...
3
votes
0
answers
130
views
Characterization of topological entropy?
Let $V$ be a smooth Anosov vector field on a compact $n$ dimensional manifold $X$. Let $D(X)$ denote the set of distance functions $d$ on $X$ that are equivalent to fixed Riemannian distance. For each ...
4
votes
1
answer
229
views
Leaves of stable foliation of holomorphic Anosov diffeomorphism
I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a ...
0
votes
0
answers
50
views
Unique poine in holonomies
Let $\Lambda$ be Axiom A for $C^{1+\gamma}$ $f$. I am reading this paper. I have a problem to undestand holonomies. The holonomy mapping
$$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...
1
vote
0
answers
88
views
Transverse measures in pseudo-Anosov diffeomorphisms
I've recently begun doing research involving pseudo-Anosov diffeomorphisms, which are diffeomorphisms on surfaces $f : M \to M$ admitting two singular measured foliations $(\mathcal F^s, \nu^s)$ and $(...
1
vote
0
answers
51
views
Complex differentials and measured singular foliations
I'm trying to understand the technical basis of singular foliations for pseudo-Anosov diffeomorphisms, and I've hit a bit of a strange calculation I'm having a hard time verifying/unpacking. In Fathi, ...
6
votes
2
answers
235
views
Handel's Theorem for surfaces with boundary
Handel's Theorem(Entropy and semi-conjugacy in dimension two, 1987): let $M$ denote a closed surface. Let $\vartheta$ be a pseudo-Anosov (orientation-presrv.) homeomorphism of $M$ and $g$ be an (...
6
votes
2
answers
403
views
Smooth conditional measures for strong stable foliations of Anosov flows
I am trying to prove an analytic result for gesodesic flows on negatively curved manifolds and I encountered the following dynamical-system porblem.
Let $B^n$ be $n$-dimensional balls and $h:B^{n-k}\...
4
votes
2
answers
709
views
Unstable Foliations
Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$.
Definition. $\mathcal{F}$ is said to be a unstable ...
5
votes
1
answer
411
views
Existence conditions for twisted cohomological equations?
Let $T: X \to X$ be an Anosov diffeomorphism. Suppose $f: X \to \mathbb{R}$ is Holder continuous (say with exponent $\alpha$). The question arises as to when $f$ can be written as $g \circ T - g $ ...
4
votes
2
answers
406
views
Eigenvalues for toral Anosov automorphisms
It is well known that on every $d$-dimensional torus there exists linear Anosov automorphisms.
My question is the following:
Given $k< d$ does there exists a linear Anosov automorphism of $\...
3
votes
1
answer
282
views
Accumulation points of the Birkhoff average of $m$
Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by
----------$f^km(A):=m(...
6
votes
3
answers
2k
views
When is an Anosov diffeomorphism mixing?
Let $M$ be a compact Riemannian manifold and let $T : M \rightarrow M$ be Anosov. I have read here that it is an open problem to prove that $T$ is topologically mixing if $M$ is connected. Katok and ...
12
votes
0
answers
520
views
Codimension 2 foliations on simply connected 4-manifolds
Are there examples of codimension 2 foliations on simply connected compact 4-manifolds such that
Every leaf is diffeomorphic to $\mathbb R^2$
Every leaf is dense?
Same question for 5-manifolds and ...
8
votes
2
answers
1k
views
Kalinin's formulation of the Anosov closing lemma
I'm trying to read a paper of Boris Kalinin on the cohomology of dynamical systems for a project. The material is geared towards topologically transitive Anosov diffeomorphisms (which is how the ...
5
votes
1
answer
594
views
Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle....