All Questions
Tagged with ds.dynamical-systems ag.algebraic-geometry
22 questions with no upvoted or accepted answers
11
votes
0
answers
252
views
Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?
Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
- 1,914
8
votes
0
answers
196
views
Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?
I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...
- 113
6
votes
0
answers
537
views
Counting limit cycles via curvature in Riemannian geometry
In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...
- 356
6
votes
0
answers
450
views
Differential equation of line tangent to caustics
This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...
- 281
5
votes
0
answers
140
views
Algebraic independence of limit cycles of Lienard equation
It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle.
According to this fact, we search for a related ...
- 356
5
votes
0
answers
216
views
Dynamical Mordell-Lang on Kahler manifolds?
Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...
user47305
5
votes
0
answers
303
views
Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties
The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
- 1,389
4
votes
0
answers
339
views
Why are critical points important for dynamical systems?
I have just started reading a little about (arithmetic) dynamics and it seems like critical points are very important - for instance, rational maps so that critical points have finite forward orbit (...
- 7,746
4
votes
0
answers
335
views
Algebraic Dynamics over separated schemes
I have a few questions regarding the current status of research on algebraic dynamics over separated schemes. In what follows $\varphi:X\rightarrow X$ will be a finite self-morphism of a noetherian ...
- 3,101
3
votes
0
answers
213
views
Dynamical degree and spectral radius
Let $X$ be a smooth, projective surface over an algebraically closed field $k$ of characteristic zero, and let $f \in \mathrm{Bir}(X)$ a birational map.
Let's denote $f_{\ast} : \mathrm{NS}(X) \...
- 1,325
3
votes
0
answers
141
views
Which is the number of independent components of a flat spin connection in a 4 dimension Weitzenböck spacetime?
A spin connection $A_{ab\mu}=-A_{ba\mu}$ has 24 components. The number of independent components for a flat spin connection can be counted by subtracting the constrains set by the condition of null ...
- 51
3
votes
0
answers
143
views
What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?
The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).
Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study of ...
- 111
3
votes
0
answers
88
views
Question about a length inequality in algebraic dynamics
Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of $\mathcal{O}_X$-...
- 3,101
2
votes
0
answers
170
views
Symplectic structure on moduli space of holomorphic Abelian differentials
I've heard a "symplectic structure" referred to on the moduli space of holomorphic Abelian differentials by numerous people / sources. I do not know how to interpret this - I am looking for ...
- 31
2
votes
0
answers
101
views
Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action
My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...
- 1,331
1
vote
0
answers
72
views
Equivalence between smoothly regular and analytically regular
I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...
- 803
1
vote
0
answers
92
views
Computing algebraic entropy
Could you recommend any reference for computing algebraic entropy?
Here algebraic entropy is defiened as $\lim_{n \to \infty}\log (deg (f^n))^{1/n}$ for a rational map $f $.
I saw that there are ...
- 663
1
vote
0
answers
403
views
Weakened jacobian conjecture for entire functions
A rudin's theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials.
The jacobian ...
- 2,106
1
vote
0
answers
242
views
How Markus–Yamabe implies Jacobian ?
To make myself precise, I would like to recall some backgrounds.
(Markus-Yamabe, $\mathrm{MY}_n$) Given a $C^1$ map $f:\Bbb R^n \to \Bbb R^n$ with $f(0)=0$ and $Df$ everywhere Hurwitz stable (the ...
- 355
1
vote
0
answers
225
views
Functions holomorphic on a region minus a Cantor set - pt.2: Iterated function systems
This post is a follow up to my previous question enquiring whether it is always possible to extend a homeomorphism conformal on a region $R$ minus a Cantor set to the whole of $R$. From the answers I ...
- 367
0
votes
0
answers
70
views
Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
- 247
0
votes
0
answers
261
views
Is an immersed Kronecker join always a multilinear variety on a Hilbert space?
The question asked is:
Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space?
This is related to another MathOverflow question
In ...
- 1,389