**0**

votes

**1**answer

78 views

### Ergodic automorphisms of a compact metric abelian group are Bernoulli

In the literature, such as in this article, it is proved that every ergodic automorphism of a compact metric abelian group is Bernoulli. A rotation is not isomorphic to a Bernoulli shift because it ...

**16**

votes

**4**answers

795 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 ...

**0**

votes

**0**answers

81 views

### Flow on invariant Lagrangian tori

The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

**2**

votes

**0**answers

53 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 ...

**10**

votes

**1**answer

368 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**1**

vote

**1**answer

74 views

### Definition o branched 1-manifold [closed]

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia:
"A finite graph whose edges are smoothly embedded arcs in a surface, ...

**1**

vote

**0**answers

183 views

### A Perron-Frobenius problem

Let $A$ be an irreducible nonnegative matrix with column sums equal to 1.
Let $b\in R^n$ have components summing to 0, and let $u$ be the solution of $u=Au+b$ with components summing to 1 (unique ...

**3**

votes

**2**answers

1k 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 ...

**2**

votes

**1**answer

31 views

### Difference between Hopf-Turing bifurcation and traveling-waves bifurcation in reaction-diffusion systems

I am perturbing a dynamical system with a perturbation that looks like $$\vec u e^{-i k x} e^{\sigma(p;k)t}$$ where $\sigma$ is a function of the parameters of the dynamical system and the wavenumber, ...

**6**

votes

**0**answers

92 views

### cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...

**7**

votes

**1**answer

281 views

### electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...

**4**

votes

**1**answer

132 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 ...

**-1**

votes

**1**answer

81 views

### Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...

**4**

votes

**1**answer

51 views

### Problem on differential inclusion

For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the ...

**17**

votes

**3**answers

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 ...

**2**

votes

**0**answers

137 views

### Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...

**3**

votes

**1**answer

150 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 ...

**23**

votes

**4**answers

1k views

### Stability of the Solar System

My question is simple:
Is the Solar System stable?
You can see this Wikipedia page.
In May 2015 i was in the conference of Cedric Villani at Sharif university of technology with this title: ...

**1**

vote

**2**answers

109 views

### Do ergodicity, minimality and equicontinuity on a compact space imply total ergodicity?

Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ?
According to some results I found in some books, a rotation on a ...

**7**

votes

**0**answers

103 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 ...

**1**

vote

**0**answers

50 views

### Ordering periodic orbits

I want to prove the proposition:
Proposition- Let $f:I \to I$ be continuos, and let f have a (2n+1)- periodic orbit {$x_{k}=f^{k}(x_{0})$, $k=0,1,\dots,2n$}, but no (2m+1)-periodic orbit for ...

**2**

votes

**1**answer

85 views

### Multi dimensional symbolic dynamics

I want to learn Multi dimensional symbolic dynamics. can you point to any recent thesis containing a good exposition or lecture notes?

**4**

votes

**1**answer

183 views

### The converse of von Neumann's mean ergodic theorem

Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following.
Let $\{F_n\}_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ ...

**38**

votes

**5**answers

2k views

### Which polynomial's roots are its coefficients?

Start with any polynomial of degree $n$ with complex coefficients, e.g.,
$$z^3+z^2+2 z+3 \;.$$
Find its $n$ roots, and list them in order of their modulus:
$$-1.28, (0.14\pm 1.53 i)$$
Now form a new ...

**4**

votes

**0**answers

214 views

### The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...

**1**

vote

**0**answers

41 views

### Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?

I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...

**5**

votes

**1**answer

96 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) ...

**21**

votes

**1**answer

1k views

### Anti-Mandelbrot set

I clearly remember seeing a paper where the dynamic of the anti-conformal map
$f(z)=\overline{z}^2+c$ was studied (the bar means complex conjugation). There was a picture of the analog of the ...

**1**

vote

**1**answer

83 views

### Convergence to equilibrium via gradient descent

J. B. Rosen proved that in concave games of n players (which assumes that Cartesian product of strategy profiles is convex) if the game satisfies the condition of diagonally strictly concave then ...

**3**

votes

**0**answers

79 views

### Perturbations to a vector field

I ran into some problems while working through a proof of the Poincare-Hopf theorem that essentially boiled down to the following question: given a smooth vector field $V$ on a (compact Riemannian) ...

**2**

votes

**0**answers

44 views

### When do positively invariant subset contain a given set?

Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't ...

**1**

vote

**0**answers

105 views

### Discrete group action on the sphere

Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$
be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...

**4**

votes

**1**answer

95 views

### When is the time one map of a suspension flow ergodic?

I'm sure the answer to the following question is well known but I couldn't find the answer I needed.
Let $(\Sigma,\sigma)$ denote the full shift on $k$ symbols and let $\mu$ be an invariant measure ...

**3**

votes

**2**answers

78 views

### Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...

**4**

votes

**1**answer

114 views

### Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...

**10**

votes

**3**answers

620 views

### Dynamical systems with multidimensional, complex and other exotic kinds of time spaces

As one may know, a dynamical system can be defined with a monoid or a group action on a set, usually a manifold or similar kind of space with extra structure, which is called the phase space or state ...

**0**

votes

**1**answer

79 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

**3**

votes

**2**answers

529 views

### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**8**

votes

**3**answers

197 views

### Uniquely ergodicity and polynomial ergodic average

Let $(X,T)$ be a uniquely ergodic system (here X is compact, T is a continuous map form $X$ to itself), so for any continuous function $f:X\rightarrow\mathbb{R}$ we have for any $x\in X$, the ergodic ...

**12**

votes

**2**answers

433 views

### Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...

**6**

votes

**1**answer

136 views

### Are irrational multiples of central sets again central?

Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact metric space, $T$ a ...

**6**

votes

**1**answer

161 views

### Is there a universal $\omega$-limit set?

For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$.
For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...

**3**

votes

**1**answer

150 views

### Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.
For any $x \in E$, there is a ...

**14**

votes

**1**answer

552 views

### A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ...
I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...

**3**

votes

**3**answers

112 views

### Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} ...

**3**

votes

**1**answer

132 views

### Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...

**5**

votes

**0**answers

660 views

### A question on “The weakened Hilbert 16th problem”

In this question we are interested in the number of limit cycles which appear in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
...

**1**

vote

**0**answers

44 views

### Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...

**7**

votes

**2**answers

196 views

### Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...

**1**

vote

**0**answers

82 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...