**4**

votes

**1**answer

190 views

### Reference request: Invariant sets of dynamical systems

(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...

**2**

votes

**0**answers

52 views

### Steady state of a dynamical equation

Suppose we have the following dynamical equation:
$P(k+1)=A\bigg(P(k) - P(k)H^T(k)\big(H(k)P(k)H^T(k)+Z\big)^{-1}H(k)P(k)\bigg)A^T+W$
with $P(0)=0$, where $P$, $A$, $H$, $Z$, $W$ are all $N\times N$ ...

**2**

votes

**0**answers

80 views

### Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...

**-4**

votes

**1**answer

142 views

### A question regarding Kingman's theorem as described in “Ergodic Theorems” book written by Ulrich Krengel [closed]

Hi so I am reading Ulrich Krengel's book called Ergodic Theorems.
And in the proof of Kingman's theorem I don't understand why on page 48 $$d_n(1,1)\leq
d_n(r,r)^{-1/(r-1)}$$, how did he get this ...

**1**

vote

**1**answer

186 views

### Algebraic topology, Dynamical systems [closed]

Let $T^2$ be a 2-torus and $f:T^2\rightarrow T^2$ a smooth map. Let $f_*:\pi_1(T^2)\rightarrow\pi_1(T^2)$ be the induced map on the fundamental group $\pi_1$. If $f_*$ has no eigenvalue greater than ...

**1**

vote

**0**answers

92 views

### The value of the sequence generated by the substitution

Given a substitution $1\to 100$, $0\to 01$, then we have $1\to 100\to1000101\to10001010110001100\to\cdots$, we denote this limits (fixed point of this substitution) as $(a_n)$, given $\beta>1 $. ...

**0**

votes

**0**answers

91 views

### Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...

**1**

vote

**0**answers

126 views

### The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...

**2**

votes

**1**answer

135 views

### Weak Convergence to Lebesgue Measure

I'm trying to understand the proof given by D. Rudolph in his paper "x2 and x3 invariant measures and entropy". I'm particularly trying to undestand the proof of lema 4.4.
Let's consider a secuence ...

**5**

votes

**0**answers

111 views

### Growth in families of trees

I'm hoping that the question below is simple thermodynamic formalism, but I can't quite make it work. Any help would be very welcome.
Let $\Sigma:=\{0,1\}^{\mathbb N}$ and let $\Sigma^*$ be the set ...

**2**

votes

**1**answer

306 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 ...

**3**

votes

**0**answers

140 views

### A special non vanishing vector field on $S^{3}$

Is there a non vanishing vector field on $S^{3}$ with an infinite family $T_{\lambda}$ of invariant torus such that each $T_{\lambda}$ has an structure of a Kronecker foliation but for ...

**3**

votes

**1**answer

128 views

### Substitutions and Sturmian sequences

We know that any substitution can generate sequence, for example the Fibonacci substitution:
$\sigma(0)=01, \sigma(1)=0$, then we can define a Sturmian sequence $\omega$, i.e., the fixed point of ...

**0**

votes

**0**answers

196 views

### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...

**2**

votes

**2**answers

187 views

### A question about transitivity

Recently in something that I'm studying, I needed to know if the following map is transitive: $\sigma: M^{\mathbb{N}}\to M^{\mathbb{N}}$ the unilateral shift, where $M$ is a uncountable compact metric ...

**3**

votes

**2**answers

187 views

### Isomophism between two irrational rotations

Define two dynamical systems $([0,1), \mathbb{B}_1, \mathbb{L}, T_1(x)=x+\alpha_1\mod 1)$ and $([0,1), \mathbb{B}_2, \mathbb{L}, T_2(x)=x+\alpha_2\mod 1)$, where $\alpha_1,\alpha_2 $ are two ...

**31**

votes

**2**answers

1k views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**6**

votes

**2**answers

190 views

### Geodesic flow on infinite surfaces

The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ...

**1**

vote

**0**answers

59 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:
...

**2**

votes

**2**answers

126 views

### Question on the number of equilibria

Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or ...

**4**

votes

**2**answers

253 views

### Lecture notes on semi group theory for linear evolution equations

I am reading (or trying to read :)) One parameter semigroups for Linear Evolution equations by Klaus-Jochen Engel and Rainer Nagel. I was wondering if someone was aware of a good set of lecture notes ...

**0**

votes

**0**answers

97 views

### Periodicities of a Complex Dynamical System

Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number.
It is easy to ...

**3**

votes

**0**answers

93 views

### Conjectural growth rate for ergodic sums of logarithms

Let $\theta, \phi \in [0,1)$, and consider the sums
$$
S_n(\theta,\phi)=\sum_{k=0}^n \log|e^{2\pi i (k\theta+\phi)}-1|.
$$
The possible boundedness from above of such sums plays a key role in ...

**3**

votes

**1**answer

164 views

### Self-diffeomorphisms of $\mathbb{R}^2$

I guess my question is very vague. Let $\phi: \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation-preserving diffeomorphism that fixes the origin. Does $\phi$ take any sort of "normal form" if we allow ...

**3**

votes

**0**answers

58 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

**2**answers

108 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
...

**2**

votes

**1**answer

148 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

**1**answer

432 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

**1**answer

286 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

**0**answers

81 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

**2**answers

124 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 ...

**6**

votes

**2**answers

151 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

**3**answers

317 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 ...

**2**

votes

**0**answers

154 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.$g_{*}X=\pm X$ where $g$ ...

**2**

votes

**0**answers

111 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

**1**answer

98 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

**0**answers

129 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 ...

**6**

votes

**1**answer

151 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

**0**answers

113 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

**0**answers

73 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

**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 ...

**44**

votes

**5**answers

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

**0**answers

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

**0**answers

102 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

**1**answer

266 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

**0**answers

456 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

**3**answers

1k 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 ...

**16**

votes

**2**answers

864 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

**1**answer

118 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

**2**answers

214 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 $\chi^{\infty}(M)$ ...