**7**

votes

**1**answer

233 views

### Analytic diffeomorphisms of the circle from complex domains

Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms:
$$\phi : (D^2,S^1) \to ...

**4**

votes

**0**answers

284 views

### A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...

**4**

votes

**0**answers

117 views

### Unit eigenvalue of the linearized Poincare return map

Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...

**5**

votes

**0**answers

182 views

### Is Akcoglu's theorem for power bounded positive operators still an open problem?

I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.
" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...

**3**

votes

**1**answer

147 views

### Is there literature available on iterated function systems of the form $f^n = (g f^{n - 1}, g f^{n - 2}, \ldots)$?

This question is motivated by another question on math.stackexchange.
From a function $g:X^k\to X$ it is possible to define an iterated function system on $X^k$ with the function $f:X^k\to X^k$ ...

**0**

votes

**1**answer

186 views

### Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

**3**

votes

**0**answers

170 views

### Derivative of Wronskian

In the proof of Theorem 2 in this paper here on arxiv on page 10 for $k=2$ it is claimed that if the Wronskian of two solutions $y_1,y_2$ to the differential equation
$$-y''_i(x) + q(x) y_i(x) = ...

**0**

votes

**0**answers

94 views

### Asymptotic pseudo orbit of an action

Let $G$ be finitely generated group (i.e $G= <S>$ $S=\{ s_1, ...,
s_n\}$) and $\varphi:G\times M\longrightarrow M$ is an action then
$f:G\longrightarrow M$ is called $\delta$- pseudo orbit if ...

**4**

votes

**2**answers

330 views

### Is there any elementary proof of No wandering domain for polynomials

It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps.
I think it is interesting to ask whether we ...

**14**

votes

**1**answer

347 views

### Nonperiodic points of homeomorphisms of a ball

Suppose $B$ is a $d$-dimensional ball (for some $d \geq 1$) and $T$ is a homeomorphism from $B$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that ...

**3**

votes

**1**answer

257 views

### invariant measures of the expanding maps on the circle

I would be very happy to know about original references for the following results;
For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1)
(1) There exist ...

**1**

vote

**1**answer

191 views

### Vector fields whose divergence are proper maps

Let $X$ be a polynomial vector field of degree $2$ on $\mathbb{R}^{2}$. Does there exist a nonvanishing smooth function $g$ such that $Div(gX)$ is a proper map?Or at least the zero locus of ...

**1**

vote

**1**answer

93 views

### Groups arising as direct limits of a stationary system of primitive matrices over the integers

I am interested in the kinds of groups of the form $\displaystyle\lim_{\longrightarrow}(\mathbf{Z}^k,M)$ where $M$ is a primitive (some power of $M$ has strictly positive components) $k\times k$ ...

**5**

votes

**1**answer

184 views

### Non-closed geodesics on a convex polyhedron in $\mathbb{R}^3$

Let $P$ be the surface of a closed convex polyhedron in $\mathbb{R}^3$.
Q. Does every non-closed geodesic $\gamma$ fill $P$ densely?
Of course $\gamma$ cannot pass through a vertex of $P$, but ...

**10**

votes

**1**answer

319 views

### Nonperiodic points of piecewise-linear homeomorphisms

Suppose $K$ is a compact polytope and $T$ is a piecewise-linear homeomorphism from $K$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that ...

**1**

vote

**1**answer

175 views

### Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle

Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much.
Let $M$ be a n dimensional manifold. ...

**3**

votes

**0**answers

80 views

### Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...

**0**

votes

**0**answers

77 views

### Rational dynamical system with nonnegative paramaters

let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...

**9**

votes

**1**answer

329 views

### Perron-Frobenius theory for reducible matrices

Can someone suggest some sources/references dealing with the Perron-Frobenius theory for nonnegative matrices that are reducible?
Specifically, if $A\ge 0$ is a $d\times d$ matrix with no assumptions ...

**4**

votes

**1**answer

195 views

### Resolvent of a triangular matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$?
Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...

**2**

votes

**2**answers

209 views

### Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?

For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision ...

**3**

votes

**2**answers

142 views

### The upper and lower bound of the projection of a subshift of finite type

I am thinking a problem: given a subshift of finte type of $\{0,1\}^{\mathbb{N}}$ and $2>q>1$, where $q$ is a real number. Then how can we find the largest and smallest numbers of the projection ...

**12**

votes

**2**answers

439 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 $\mathbb{R}^2$ ...

**2**

votes

**1**answer

165 views

### Divergence invariant lifting of a vector field via a submersion

What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is Not true:
For every vector field $X$ on $S^{2}$ there is a non vanishing vector field ...

**2**

votes

**2**answers

373 views

### Rate of convergence of an irrational rotation

Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$.
If we assume that $\alpha$ is irrational, then there exists an increasing ...

**3**

votes

**0**answers

90 views

### Topological pressure for subshifts on a countable alphabet

Apologies for asking two similar questions within a week of each other, I had hoped that asking a finite alphabet version of this question would lead to enlightenment but unfortunately it didn't.
...

**4**

votes

**3**answers

202 views

### Approximating Subshifts From Below

I'm looking to understand how to approximate certain countable alphabet subshifts by Markov shifts, and realised that I don't know how to do it even in the finite alphabet case. My guess is that the ...

**1**

vote

**0**answers

130 views

### Properties of a function from its pullback

Edit: I have now removed the duplication previously referred to. Thank you.
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ ...

**3**

votes

**0**answers

253 views

### Classification of complex Kronecker foliations

Let $\theta \in \mathbb{C}$ be a fixed complex number. The submersion $f:\mathbb{C}^{2}\to \mathbb{C}\; \text{with}\; f(x,y)=y-\theta x$ defines a complex foliation on $\mathbb{C}^{2}$. Consider the ...

**1**

vote

**1**answer

99 views

### Is it possible to define the density of the logistic map for $x<0$?

Probability density functions (PDF's) have inherent connections to the field of
Dynamical Systems.
The motivation for this question can be found in: ...

**1**

vote

**2**answers

204 views

### Two questions related to $TS^{2}$ as a holomorphic manifold

We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. ...

**2**

votes

**0**answers

124 views

### A (different) foliation arising from Hopf fibration

In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...

**4**

votes

**1**answer

197 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

57 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

86 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

149 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

210 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

94 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

109 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

143 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

156 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

115 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 ...

**4**

votes

**2**answers

569 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 ...

**4**

votes

**1**answer

216 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

143 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 ...

**1**

vote

**0**answers

233 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

192 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

218 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 ...

**33**

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

226 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 ...