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

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7
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3answers
528 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 ...
1
vote
1answer
117 views
+50

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 $\chi^{\infty}(M)$ ...
13
votes
2answers
582 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 ...
0
votes
0answers
60 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
0answers
77 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 ...
8
votes
0answers
192 views
+100

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
0
votes
0answers
96 views
+50

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 ...
3
votes
1answer
806 views

Invariance of dynamical system under a transformation

I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out why this is happening (for quite some time now actually). Any help ...
4
votes
1answer
118 views

introduction books for Dynamic systems of discrete Schrodinger operator for beginner

In this semester, I study in a class of dynamic system. recently the French professor turn to the dynamic system of discrete operator. I find it is difficult to find a book in English. (I have found ...
3
votes
1answer
92 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
162 views

NonCommutative analog of the Poincare Bendixson theorem [on hold]

The classical Poincare Bendixson theorem says: If $X$ is a vector field on $\mathbb{R}^{2}$ or $S^{2}$ and $K$ is a compact set which is invariant under the flow generated by $X$. Then $K$ contains ...
10
votes
2answers
1k views

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 : Most interesting mathematics mistake? Added 1: According to their method, ...
0
votes
0answers
93 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 ...
19
votes
1answer
492 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 ...
4
votes
0answers
109 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 ...
0
votes
1answer
90 views

Find a sequence with uniform frequencies and recurrent property

Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$ such that this sequence has uniform ...
2
votes
0answers
196 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
3
votes
0answers
68 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 ...
4
votes
1answer
463 views

The space of diffeomorphisms on a manifold

It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions ...
5
votes
0answers
103 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
475 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
98 views

Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function, ...
-1
votes
0answers
64 views

if $X$ is a vector field in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity?

if $X$ is a vector field in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? And in dimension $n$? I know there is singularity when $n=2$.
6
votes
1answer
99 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 ...
3
votes
1answer
119 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 $U$. ...
16
votes
2answers
533 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 = ...
1
vote
1answer
88 views

Complement of bifurcation variety

I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of $A_k$, $D_k$, $E_k$ and lagrangian singularities". Let $f\colon \mathbb{C}^n\to ...
1
vote
0answers
187 views

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 ...
1
vote
1answer
203 views

Does specification implies that entropy map is upper semicontinuous?

Let (X,d) be a compact metric space and f a continuous transformation on X. f has the specification if one can always find a single orbit to interpolate between different pieces of orbits, up to a ...
2
votes
0answers
107 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 ...
40
votes
10answers
6k views

The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$ (so $c_0=0$ is imposed). First things that ...
5
votes
3answers
114 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 ...
2
votes
1answer
102 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: ...
0
votes
0answers
51 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 ...
0
votes
1answer
224 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
216 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 ...
4
votes
1answer
118 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 $T(I_i)=[0,1]$. ...
0
votes
0answers
78 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: ...
1
vote
0answers
102 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) derivation on $\mathbb{C}[x,y]$: ...
4
votes
1answer
155 views

Homeomorphisms that admit a decomposition

Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$. If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ...
3
votes
2answers
134 views

Decay of Correlation, references for a non-standard way

In ergodic theory is common to use the decay of correlation property to deduce properties analogues to those of i.i.d. random variables. Call $X\doteq [0,1].$ Examples of decay of correlation ...
5
votes
0answers
216 views

Deterministic shifts

We consider (topological) dynamical systems $(\Omega, S)$, where $S$ is the shift $(Sx)_n=x_{n+1}$, and $\Omega\subset[0,1]^{\mathbb Z}$ is a compact, shift invariant subspace. I call such a system ...
5
votes
1answer
211 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 ...
0
votes
1answer
88 views

Non-convergence of ergodic measures with positive entropy

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X$ with strictly positive Sinai entropy $h_{\mu}(T).$ Let ...
1
vote
1answer
71 views

Example of non-convergence of iteration of measures

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X.$ Let $F:X\to X$ be a continuos transformation that commutes ...
0
votes
1answer
109 views

Is the following conjecture regarding rotational iterates of collection of points on a circle true?

Let $\varphi_1,\dots,\varphi_k$ be a set of angles in $[0,\pi)$. For each $n\in\mathbb{N}$, let $$ \rho(n):=\min_{j=1,\dots,k} \{n\cdot \varphi_j \mod \pi \}. $$ Is it true that for infinitely many ...
7
votes
2answers
225 views

The importance of differentiable dynamics from outside dynamics? (mainly topology)

I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics. This could potentially include ...
4
votes
0answers
90 views

Kopell's lemma in higher dimensions

Kopell's lemma states the following: suppose f and g are commuting $C^2$ diffeomorphisms of $[0, \infty)$ such that $f(x) < x$ for every $x > 0$ and $g(p) = p$ for some $p > 0$. Then $g$ is ...
6
votes
4answers
595 views

Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$

The group $\mathrm{PSL}_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace ...
2
votes
1answer
121 views

$C^1$ stability conjecture on non-compact manifolds

In the study of dynamical systems, the link between structural stability and Smale's Axiom A has been explored by many authors. One of the important outcomes of this endeavor was the proof of $C^1$ ...