1
vote
0answers
46 views

square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces

I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant: ...
0
votes
0answers
110 views

Manifolds supporting finite order diffeomorphisms (a local construction?)

The following question is mainly inspired by this previous one Which manifolds admit a diffeomorphism of order $n$? and some answers given there. For $d\geq 2$, let $\mathbb{B}^d$ denote the closed ...
0
votes
3answers
342 views

Good books on Geometric Theory of Dynamical Systems

I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...
10
votes
2answers
411 views

Vector field on 3-sphere

Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is ...
14
votes
3answers
606 views

fixed point property for maps of compacts

Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point. Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...
4
votes
2answers
269 views

Complexification or 'real'ization of Mapping Class group.

So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
6
votes
1answer
503 views

Topological equivalence of homotopic vector fields

Two (tangent) vector fields $X$ and $Y$ on oriented differentiable manifolds $M$ and $N$, respectively, are topologically equivalent, if there is an orientation-preserving homeomorphism $M \to N$, ...
1
vote
1answer
160 views

Pseudo-Anosov map with n-prong singularity

Whether the following statemant is correct (I guess the answer is "Yes" and I guess that maybe it is trivial for an expert about Pseudo-Anosov map)? "For a given $n\in N$, there exists a closed ...
2
votes
0answers
265 views

Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them. Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...
3
votes
1answer
225 views

interval exchange maps and surfaces

I apologize if the question is a well-known theorem, but I'm just starting to learn about laminations, so I don't know much. The question is roughly, if interval exchange maps have an underlying ...
1
vote
1answer
122 views

Example of dynamical system $M$ such that $M \rightarrow \mathbf{R}\backslash M$ is not locally trivial?

Let $\Phi: M \times \mathbf{R} \rightarrow M$ be a smooth dynamical system having no periodic orbits, i.e. such that the canonical map $\pi:M \rightarrow \mathbf{R}\backslash M$ is a principal ...
2
votes
1answer
261 views

A kind of foliantion on figure eight knot complement

Let $N$ be the figure 8 knot complement, What we can say about such kind of dim 2 foliation $F$ on $N$: (1) no Reeb (2 dim); (2) $F$ intersect transversly with $\partial N$ is $n$ pareller Reeb (1 ...
1
vote
0answers
142 views

Linearization of singular foliation in the plane

Hello, I would like to obtain a smooth model around a singular point of a foliation (in my case the model should be the linearization of the foliation). It seems that the answer could be hidden in ...
4
votes
1answer
191 views

Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...
3
votes
1answer
314 views

pseudo-Anosov surface in three manifolds

A surface $S$ in a three manifold $M$ is pseudo-Anosov means if there exists a homeomorphism $f$ over $M$ for which $S$ is $f$ invariant and $f$ is a pseudo-Anosov on $S$. For example, $M$---- any ...
1
vote
0answers
140 views

Sectional surface of dimension 1 foliation on 3 manifolds

In link text, I have considered a kind of sectional surfaces (i.e., regular level sets) about a spectial type of nonsingular flows (i.e., Smale flow) on three manifolds. I obtained that a 3-manifold ...
2
votes
1answer
558 views

How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?

What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
14
votes
3answers
641 views

Codimension zero immersions

Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk? Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...
2
votes
0answers
126 views

System dynamic of space euclidean and hyperbolic tilings

Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC) tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of $R^{d}$ by translation is on ...
7
votes
2answers
553 views

Glue two solenoids along their boundaries

Here a solenoid is a dynamical system $(N,f)$ where $N$ is the solid torus $N=\mathbb{D}^2\times S^1$ with boundary $S^1\times S^1$, and $f:N\to N$ is a smooth embedding whose image is wrapped twice ...
4
votes
5answers
2k views

nowhere vanishing vector field on a manifold

I am wondering if there are necessary and sufficient conditions under which an one-dimensional subbundle of $TM$ has a nowhere vanishing vector field. More precisely let $M$ be a compact smooth ...
3
votes
3answers
591 views

Geometry and Integrability in Other Bundles

Background: Suppose $E=TM$ is the tangent bundle to some differentiable manifold $M^n$. If we specify some subbundle $D\subset TM$ (distribution of $k$-planes) then there are two natural situations ...
4
votes
3answers
732 views

How can one express the Dedekind eta function as a sum over the lattice?

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors ...
3
votes
0answers
598 views

Entropy conjecture for flows

The entropy conjecture for diffeomorphisms (see for example this paper) asserts that for diffeomorphisms of manifolds, the log of the spectral radius of the actions of the diffeomorphism on the ...
26
votes
3answers
2k views

Is there a reset sequence?

There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone ...
94
votes
1answer
9k views

What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
11
votes
3answers
2k views

Permute Wada Lakes keeping the coastline intact? (still open in dim >2)

Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical ...
5
votes
1answer
345 views

Automorphisms of $\pi_1$ induced by pseudo-Anosov maps

Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$. For any ...
3
votes
1answer
689 views

How to shown that the Tangent Bundle of a vector space is a Vector Bundle

Hello, I have the following question about the tangent bundle $T_M = \bigcup_{p \in M} \{p\} \times T_p M$ defined on a manifold $M$ of class $C^r$ modeled on a normed space $X$. My problem is ...
5
votes
5answers
1k views

How can generic closed geodesics on surfaces of negative curvature be constructed?

As far as I understand it the closing lemma implies that closed geodesics on surfaces of negative curvature are dense. So: how can they be constructed in general? A concrete answer that dovetails ...
11
votes
3answers
2k views

Mapping Class Groups of Punctured Surfaces (and maybe Billiards)

Where can I find a concrete description of mapping class group of surfaces? I know the mapping class group of the torus is $SL(2, \mathbb{Z})$. Perhaps, there is a simple description for the sphere ...
18
votes
0answers
670 views

Almost complex 4-manifolds with a “holomorphic” vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$? The following sub question is ...
4
votes
2answers
1k views

Proper families for Anosov flows

So I've been skimming Bowen's 1972 paper "Symbolic Dynamics for Hyperbolic Flows" hoping it would give me some insight into how to build a Markov family for the cat flow (i.e., the Anosov flow ...