Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations.

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3
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0answers
51 views

Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing

The following result is on page 26 of this paper by Ferenczi [PDF]. Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., $\mu(T^nA \cap B) ...
2
votes
2answers
98 views

Making a system of second-order ODEs chaotic

Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions $X = (x_1, x_2, ..., x_N)$, we have ...
1
vote
1answer
117 views

Smoothness in Ecalle's method for fractional iterates

Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is $f(f(x)) = \sin x,$ which is real analytic for $0 < x < \pi$ but ...
5
votes
1answer
394 views

Differential Geometric Aspects of Rubber Bands

What happens, if a rubber band ( of length $l_0$ that has been stretched to length $l_1:=l_0+\Delta l\;$ and brought into the shape of a closed curve in $\mathbb{R}^3$ ) is released and if the only ...
10
votes
1answer
281 views

Trapping a particle

A particle starts a brownian walk in the middle of a long tunnel in the plane, at one end of the tunnel is a region Y of given area A. Does the shape of region Y affect average time for the particle ...
1
vote
0answers
62 views

A question on periodic points and recurrent points

I have been reading the book "dynamical systems and semisimple groups an introduction". In this book, a point of a topological $G$-space $X$ is a periodic point if $G/G_x$ is compact, where $G$ is a ...
2
votes
2answers
92 views

Complementary integrable vector fields

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric ...
5
votes
2answers
141 views

Algorithm for determining when polynomial iteration is bounded?

Let $f: \mathbb{Q}\to \mathbb{Q}$ be a polynomial map with rational coefficients. Let $p\in \mathbb{Q}^n$. Is there a known algorithm that given this data determines whether or not the iterates ...
4
votes
3answers
300 views

Generic absence of non-trivial first integrals of geodesic flows

Is it true that given a smooth manifold M (with or without boundary), a "generic" metric g on M does not possess any non-trivial (non-constant) first integral for the geodesic flow induced by g on the ...
0
votes
0answers
62 views

Characterization of certain analytic vector fields on $S^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies: $X$ has a finite number of singularities on $S^{2}$ The equator is invariant under flow of $X$ 3.$g_{*}X=X\; ...
2
votes
0answers
58 views

Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin?

Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves. Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of ...
7
votes
1answer
92 views

Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?

Given a planar triangulation of (say) a convex region, imagine the following process to convert it to a triangulation with no obtuse angles: Pick an arbitrary obtuse angle at vertex $a$ of ...
1
vote
0answers
108 views

Why Poincare sphere compactification and not torus compactification

The Poincare compactification is a method to carry a polynomial vector field on the plane to an analytic vector field on $S^{2}$ via analytic embedding $$(x,y)\to ...
5
votes
1answer
117 views

“The” kronecker foliation or “a” kronecker foliation?

Consider the following two foliations of torus: 1)The Kronecker foliation with slope $\sqrt{2}$ 2)The Kronecker foliation with slope $\pi$ As I learn from the literature, these two foliations are ...
1
vote
0answers
109 views

The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$ It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...
1
vote
0answers
68 views

Smooth normal forms of vector fields (the path method)

I start by considering a polynomial vector field $$F=\varepsilon\frac{\partial}{\partial x}-(z^2+x)\frac{\partial}{\partial z}+0\frac{\partial}{\partial \varepsilon}.$$ Next I define a perturbation of ...
14
votes
3answers
1k views

Prime factorization “demoted” leads to function whose fixed points are primes?

Let $n$ be a natural number whose prime factorization is $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$ Define a function $g(n)$ as follows $$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \,$$ i.e., exponentiation is ...
44
votes
5answers
3k views

Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...
0
votes
0answers
48 views

Involution on $\chi^{\infty}(\mathbb{R}^{n})$

Is there a Lie algebraic involution $\theta$ on $\chi^{\infty}(\mathbb{R}^{n})$ which restriction to linear vector fields is the involution $\theta(A)=-A^{tr}$ for $A\in M_{n}(\mathbb{R})$? ...
4
votes
0answers
99 views

First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...
11
votes
1answer
257 views

A strengthening of base 2 Fermat pseudoprime

If $n$ is a prime then for all $k$ with $1 \le k \le [n/2]$, $k$ divides ${n-1 \choose 2k-1}$ because of the identity ${n-1 \choose 2k-1} \frac{n}{k}=2{n \choose 2k}$. My question is whether an ...
15
votes
0answers
433 views

The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with ...
14
votes
3answers
968 views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
15
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2answers
794 views

If there is a dense geodesic, are almost all geodesics equidistributed? Dense?

Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...
1
vote
1answer
109 views

Generic path in the space of vector fields on the orientable surface

Who knows, does a theorem of such form exist: Any one-parameter family of vector fields on the orientable surface may be slightly perturbed such that 1) all fields in the family except finite number ...
1
vote
2answers
203 views

A question on involutions on the Lie algebra of vector fields

Edite According to the essential comment of Ian Agol I revise the question as follows For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ ...
1
vote
0answers
96 views

Positively invariant set

In my research I need to show that the set $$S:=\left\{x_{i}\in [0,1] , y_{i} \in [0,c_{i}] ; i = 1,\dots , n; \sum_{i=1}^{n}(x_{i}+y_{i}) =1\right\},$$ is positively invariant with respect the ...
3
votes
1answer
144 views

Are there $0$ entropy non-atomic invariant measures for $2x$ and $3x$ modulo $1?$

This question appears for first time (to my knowledge) in ×2 and ×3 invariant measures and entropy Daniel J. Rudolph Ergodic Theory and Dynamical Systems / Volume 10 / Issue 02 / June 1990, pp 395 - ...
0
votes
0answers
186 views

A derivational approach to the Poincare Bendixson Theorem

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a smooth vector field on the plane. Assume that $K\subset \mathbb{R}^{2}$ is a compact subset (not necessarily invariant under ...
1
vote
0answers
110 views

Name/terminology for a relationship between group actions

Let $G$ and $H$ be groups, both acting on a set $X$. Suppose that there is a homomorphism $\phi:G\to H$ such that for every $g\in G$ and $x\in X$, $g\cdot x = \phi(g)\cdot x$. Is there a name for this ...
6
votes
0answers
139 views

Measure theoretic entropy

I don't know if this is an elementary question or not. In what follows all maps are continuous Suppose that $P:\mathbb{C}\rightarrow\mathbb{C}$ is a complex polynomial of degree $d>1$ and let ...
23
votes
2answers
669 views

Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$. There are conditions on $\{ p_1, p_2, d \}$ for this ...
6
votes
0answers
148 views

The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow ...
18
votes
0answers
532 views

Base change for $\sqrt{2}.$

This is a direct follow-up to Conjecture on irrational algebraic numbers. Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...
4
votes
1answer
130 views

Jacobian has small eigenvalues everywhere in an open set. Does this imply globally attracting fixed point in that set?

Let $U$ be an open subset of $\mathbb{R}^2$. For my purposes, we can assume that $U$ is just a rectangle. I have an infinitely differentiable map $M:U\to U$ that has a unique fixed point $p$ in $U$. ...
6
votes
1answer
109 views

Angles and proportions occurring in L-system fractals

This is about properties of certain fractals defined by Lindenmayer systems, a.k.a. L-systems. Unlike “classical” fractals like Julia sets or the Mandelbrot set (the name “set” says it all), these ...
18
votes
2answers
724 views

Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...
2
votes
0answers
403 views
+50

The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
3
votes
0answers
117 views

On the volume entropy of negatively curved manifolds

Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the ...
2
votes
1answer
146 views

On the Birkhoff ergodic theorem for geodesic flows

Let $S$ be a closed surface endowed with a Riemannian metric of negative curvature and let $US$ be the unit tangent bundle. Let $\mu$ be the Liouville measure on $US$. Let $f: ...
6
votes
3answers
158 views

Is there a effective computational criterion to all periodic points of a rational function are repelling.

I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a ...
0
votes
0answers
58 views

Does the number of zero eigenvalues correspond to the dimension of equilibrium manifold for nonlinear system?

Consider nonlinear dynamical system with $m \times m$ Jacobian matrix $J(x)$ that has $k$ zero eigenvalues for all $x$. The rest $m-k$ eigenvalues have negative real part for all $x$. Is that true ...
18
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2answers
3k views
+400

The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post. Added : According to their ...
0
votes
1answer
243 views

Heisenberg group acts on the circle

Let $H$ be a Heisenberg group, i.e. $$ H=\left\langle a,b,c |[a,b]=c,[a,c]=[b,c]=1\right\rangle. $$ $H$ acts on the circle by homeomorphism which preserves the orientation. If the rotation number of ...
-4
votes
1answer
233 views

Proof of im/possibility of constructing any fractal by iterated function systems? [closed]

Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal(object of any non-integer dimension) can be constructed by ...
0
votes
0answers
103 views

Regarding a paper relating surfaces and integrable mappings

This question has been posted on stackexchange, without answers. I'm reading the paper "A classification of two-dimensional integrable mappings and rational elliptic surfaces". I have two questions: ...
3
votes
0answers
74 views

Regarding the definition of S-flows over a category (given a monoid S)

(This question was originally directed to Simone Virili, referring to the answer http://mathoverflow.net/a/103840/2926, but could also be addressed to the greater community.) I was wondering if you ...
1
vote
0answers
132 views

The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$: ...
6
votes
1answer
139 views

Estimates of Hausdorff dimension (and its derivatives)

For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...
5
votes
1answer
232 views

“strongly mixing” action on dimers?

In Local Statistics of Lattice Dimers we study a nice familiar object, domino tilings in the plane extending out to infinity. His paper is going to discuss the frequency of various "motifs" in ...