# Tagged Questions

**5**

votes

**1**answer

98 views

### Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...

**1**

vote

**0**answers

11 views

### Differential inequalities for a strictly diagonal dominant system of linear ODEs

Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...

**1**

vote

**0**answers

238 views

+50

### 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 ...

**7**

votes

**0**answers

817 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 ...

**16**

votes

**1**answer

552 views

### Example of a quasi-Bernoulli measure which is not Gibbs?

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only.
A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$,
$$
C^{-1} ...

**2**

votes

**1**answer

164 views

### Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com
http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe
Now I ...

**4**

votes

**0**answers

50 views

### Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow

Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...

**4**

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 ...

**1**

vote

**0**answers

96 views

### What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp

First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...

**12**

votes

**1**answer

455 views

### resampling over Bowen balls

Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...

**29**

votes

**5**answers

1k views

### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...

**1**

vote

**0**answers

60 views

### Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...

**7**

votes

**2**answers

212 views

### On the density of the sequence $\{n \{n \xi \} \}_n$

I have a question that I can't manage to answer myself. It comes from some work in PDE theory, but it is related to analytic number theory.
Let us say that we have an irrational number $\xi$. The ...

**7**

votes

**3**answers

126 views

### Ball ricochetting from a plane of close-packed spheres

Suppose the lower $z \le 0$ halfspace of $\mathbb{R}^3$ is filled with a rigid close-packing of
unit-radius spheres.
(I don't think it matters much for my purposes if it is
an
FCC or an HCP ...

**6**

votes

**1**answer

83 views

### Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geosesic flow is recurrent?

Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = ...

**2**

votes

**2**answers

74 views

### Periodic orbits in the plane

Consider a vector field $F:\mathbb{R}^2\rightarrow \mathbb{R}^2$ of the following form $F(y_1,y_2)=(y_2,\mu(y_1))$, where $\mu\in\mathscr{C}^1(\mathbb{R})$ has appropriate growth so that the solutions ...

**1**

vote

**0**answers

38 views

### Hyperbolic PDE from total derivative?

Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...

**3**

votes

**3**answers

213 views

### Maximizing entropy under constraints

This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures.
Consider the one-sided shift ...

**38**

votes

**2**answers

2k views

### Can we trap light in a polygonal room?

Suppose we have a polygonal path $P$ on the plane resulting from removal of an one of a convex polygon's edges and a ray of light "coming from infinity" (that is, if we were to trace the path ...

**0**

votes

**1**answer

60 views

### Sequence of translation surfaces and length of saddle connections

A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is ...

**-4**

votes

**1**answer

110 views

### Existence and uniqueness of solutions for a system of first order PDEs [closed]

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:
A$\dfrac{\partial}{\partial ...

**12**

votes

**2**answers

383 views

### Nuclear operators/spaces and transfer operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...

**25**

votes

**2**answers

5k 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 this related post and also the following post.
Added : According to their ...

**13**

votes

**3**answers

1k views

### Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" ...

**4**

votes

**1**answer

93 views

### Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map.
My question is: are any such measures also $T^{-1}$-invariant? BYO ...

**6**

votes

**0**answers

205 views

### Topological entropy and periodic sequences of a subshift

Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$.
...

**1**

vote

**1**answer

110 views

### Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation?
$$\begin{cases}\dot{x}=y-x^{3}\\\dot ...

**0**

votes

**0**answers

38 views

### Necessity of expansiveness for existence absolutely continuous invariant measures for piecewise smooth maps of an interval

A map $\tau:[0,1]\to[0,1]$ is piecewise smooth (or $C^r$) if there is a partition of $[0,1]$ into intervals, $[0,1]=\cup I_n$, (which can be either finite or countable) such that the restriction of ...

**1**

vote

**0**answers

57 views

### Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...

**4**

votes

**0**answers

80 views

### Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul ...

**2**

votes

**0**answers

254 views

### Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...

**2**

votes

**0**answers

673 views

### What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required.
In random signal theory, this distribution is typically a ...

**7**

votes

**2**answers

123 views

### List of Bernoulli chaotic systems

Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?
I am aware that it is known for some uniformly ...

**5**

votes

**1**answer

101 views

### General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics.
I have been interested in some technical properties of the ...

**0**

votes

**0**answers

24 views

### Equivalence classes on an ordered Bratteli diagram

Let $S$ be the adic transformation preserving a probability measure $\mu$ on the set $\Gamma$ of infinite paths of a $\mathbb{N}$-graded ordered Bratteli graph.
For every $n \geq 0$ define the ...

**19**

votes

**1**answer

632 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 ...

**2**

votes

**0**answers

66 views

### Hausdorff dimension = entropy/Lyapunov exponent for the baker's map?

Let $\Sigma=\{0,1\}^{\mathbb Z}$ and let $\sigma:\Sigma\to\Sigma$ be the left shift. Then it is well known that $(\Sigma, \sigma)$ is conjugate to the baker's map $B$ of the unit square:
$$
B(x,y) = ...

**1**

vote

**1**answer

171 views

### Applications of topology to discrete dynamical systems?

I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.
I mean cases where adding a topology to the sets ...

**2**

votes

**0**answers

42 views

### Lyapunov exponents of Lorenz63 and Lorenz96 system

Can someone suggest a reference on the mathematical results (NOT numerical) on the Lyapunov exponents of Lorenz-63 and Lorenz-96 systems (or any other non-trivial system)? In particular, is it always ...

**2**

votes

**2**answers

103 views

### Invariant $\sigma$-field of a product with a weakly mixing transformation

It is known that an invertible mpt $S$ is weakly mixing if and only if $S \times T$ is ergodic for any ergodic invertible mpt $T$. Is it more generally true that the invariant $\sigma$-field of $S ...

**0**

votes

**0**answers

47 views

### parametrizations for sections of time in a flow

Let $\phi:\mathbb{R}\times X\rightarrow X$ be a flow in $X$, where $(X,d)$ is a compact metric space. Denote by $\mathcal{H}$ the set of continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that ...

**1**

vote

**0**answers

70 views

### Is there an area preserving toral endomorphism with critical point?

An endomorphism is a continuous map $f:\mathbb{T}^2 \to \mathbb{T}^2$. An conservative endomorphism is an endomorphism that is area preserving $(m(f^{-1}(U)=m(U), \forall U$ borel set and m is the ...

**2**

votes

**0**answers

220 views

### Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question:
What's the probability distribution of a deterministic signal? (functional integrals in probability theory)
Clearly my question looks at the same time fairly ...

**0**

votes

**0**answers

27 views

### A problem from Sakai's book on derivations on C(K) and differential structure on K

In his book, Operator Algebras in Dynamical Systems, at page 59 Sakai poses the following question.
Problem: Let K be a compact space and suppose that C(K) has a non-zero closed *-derivation. Then ...

**5**

votes

**1**answer

60 views

### Survivor sets for expanding maps of the interval

Let $T:[0,1]\to [0,1]$ be a piecewise smooth expanding map, i.e., $|T'(x)|>1$ for all $x$. Let $I_n$ be a sequence of nested intervals (i.e., $I_{n+1}\subset I_n$) such that the length of $I_n$ ...

**2**

votes

**0**answers

26 views

### A proper class for smooth chaotic function

This might be a little, soft, but I'll try
Consider the interval $I=[-1,1]$. We will define a chaotic function $f:\mathbb{R}_+ \times I \to \mathbb{C}$ in the following traditional way:
For every ...

**26**

votes

**4**answers

1k views

### Stability of the Solar System

Is the Solar System stable?
You can see this Wikipedia page.
In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and ...

**0**

votes

**1**answer

53 views

### What is the relationship between solutions for the parameterised second order differential equations

Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for
$$
u''+u'-i\lambda V(x)u=0, \, x\in [0,1],
$$
What is the ...

**0**

votes

**1**answer

55 views

### Long term behavior of a certain discrete time dynamical system on graphs

Consider the graph $(V,E)$ with vertex set $V=\{v_1,...,v_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v_i\in V, (v_i,v_i)\in E$.
Assume that each vertex has an ...

**1**

vote

**0**answers

84 views

### Ergodic skew product on $\mathbb T^d\times U(2)$

Let $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ be the $d$-dimensional torus with normalized Haar measure $\mu_1$ and let $U(2)$ be the group of $2\times2$ unitary matrices with normalized Haar measure ...