# Questions tagged [anosov-systems]

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17
questions

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### 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**

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114 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 ...

**4**

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**1**answer

122 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 ...

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37 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) \...

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51 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 $(...

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42 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

165 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 (...

**7**

votes

**2**answers

294 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

559 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 ...

**4**

votes

**2**answers

365 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

252 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(...

**4**

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**1**answer

411 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 ...

**11**

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483 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 ...

**5**

votes

**1**answer

348 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 $ ...

**8**

votes

**2**answers

776 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 ...

**6**

votes

**3**answers

1k 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 ...

**5**

votes

**1**answer

548 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....