Questions tagged [anosov-systems]

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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 $C^{1, \alpha}$. ...
Guangqiu Liang's user avatar
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?
Uagi's user avatar
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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 ...
utterly confused's user avatar
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 ...
Chi Cheuk Tsang's user avatar
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 ...
Alex K's user avatar
  • 143
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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) \...
Adam's user avatar
  • 1,001
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 $(...
D. Ford's user avatar
  • 151
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, ...
D. Ford's user avatar
  • 151
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 (...
Arnaud Maret's user avatar
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}\...
twch's user avatar
  • 126
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 ...
Pengfei's user avatar
  • 2,184
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 $\...
rpotrie's user avatar
  • 3,878
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(...
Pengfei's user avatar
  • 2,184
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 ...
Steve Huntsman's user avatar
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 ...
Andrey Gogolev's user avatar
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 $ ...
Akhil Mathew's user avatar
  • 25.3k
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 ...
Akhil Mathew's user avatar
  • 25.3k
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 ...
Steve Huntsman's user avatar
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....
Andrey Gogolev's user avatar