# Tagged Questions

**5**

votes

**0**answers

88 views

### The autonomous diameter of the group of Hamiltonian diffeomorphisms of the standard symplectic space

The autonomous norm of a Hamiltonian diffeomorphism $h$ of a symplectic manifold $(M,\omega)$ is the smallest number $n\in \mathbf N$ such that $h=a_1\dots a_n$, where $a_i$ are autonomous ...

**2**

votes

**0**answers

69 views

### Question about martin boundaries of random walks induced on transient subgroups

Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
...

**1**

vote

**1**answer

59 views

### Does a continuous function have a continuous integral function in a discrete dynamical system?

Let $X$ be a compact manifold (or the closure of a Euclidean domain if that helps significantly) and $T\colon X\to X$ a homeomorphism.
Let us say that a function $v\colon X\to\mathbb R$ is the ...

**0**

votes

**1**answer

55 views

### underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form:
...

**4**

votes

**0**answers

49 views

### Why can every twist map be realized as the time-1 map of a time-dependent Hamiltonian?

if have problems getting my head around the following claim made
by Moser in "Monotone twist mappings and the calculus of variations" and Gole in "Symplectic twist maps".
Setting:
Let $F : ...

**2**

votes

**0**answers

176 views

### Mixed up by definitions of mildly mixing

Here are two setup where the notion of "mildly mixing" comes up: for representations and for group acting by measure preserving transformations (see definitions below). Since a natural class of ...

**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 ...

**6**

votes

**1**answer

117 views

### Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set?

Let $M$ be a compact Riemannian manifold, $f: M \to M$ a diffeomorphism, and $\mu$ an ergodic measure for $M$. Suppose that the support of $\mu$ is not a finite set. Is it possible that all the ...

**0**

votes

**1**answer

34 views

### How to quantify the closeness/distinctness of the attractors?

Suppose I have two discrete long-time series from two dynamical systems. I assume these two systems have compact attractor. How do I measure the closeness or distinctness of the two attractors from ...

**5**

votes

**2**answers

291 views

### Can a smooth diffeomorphisms of a Riemannian manifold have only positive Lyapunov exponents?

Let $\mu$ be some ergodic measure of our compact Riemannian manifold $M$, which is preserved by $f\in Diff^{1+\beta}(M)$. Is it possible that all the Lyapunov exponents of $\mu$ will be positive? ...

**9**

votes

**2**answers

222 views

### Two elementary inequalities for real-valued polynomials

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly ...

**7**

votes

**1**answer

87 views

### Integer Recursion Reference Request

I've run across the following recursion which at times seems very steady and predictable and at other times seems very chaotic.
Let $c_1, \dots c_k, b_0, m \in \mathbb{Z}$ with $b_0>m\ge 3$ and ...

**3**

votes

**2**answers

133 views

### Convex combinations of Bernoulli Measures

How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?
I mean, we are in the symbolic space ...

**2**

votes

**0**answers

676 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 ...

**12**

votes

**1**answer

146 views

### Applications of the Central Limit Theorem in dynamical systems

There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one:
has a ...

**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 ...

**19**

votes

**1**answer

300 views

### Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$

I made a throwaway comment on math stackexchange the other day that got me thinking about the following question. Let
$$ f_n (\alpha) = \frac1n \sum_{k=0}^{n-1} e(2^k\alpha),$$
where $e(x) = ...

**1**

vote

**0**answers

132 views

### A weighted ergodic average

According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...

**7**

votes

**0**answers

91 views

### Volume preserving conjugacy in Hartman-Grobman theorem

According to Hartman-Grobman theorem, a $C^1$ germ of diffeomorphism $f$ on $\mathbb{R}^n$ at a fixed point $x$ whose differential $Df(x)$ is hyperbolic is always $C^0$-conjugated to its differential, ...

**1**

vote

**1**answer

85 views

### Ergodicity of elementary symmetric polynomials with noncommutable variables

Let $\{X_n\}$ be an ergodic sequence of random variables, $X_n:(\Omega,\mathcal{F})\to (S,\mathcal{S})$ where the target set $S$ is a matrix ring. My question is,
Can the following limit be found ...

**1**

vote

**1**answer

76 views

### Identifying attractors in high dimensional dynamical sytems [closed]

I have a high dimensional dynamical system, and I was wondering if there is a method to identify the various attractors of the system i.e, a way of mapping the energy landscape?
I was thinking of a ...

**0**

votes

**0**answers

48 views

### Topological expansiveness map

Can anyone tell me the details about the topological expansiveness map in topological dynamical system?
I know that for a continuous map $T$, expansive means there exists an $\epsilon>0$ such ...

**9**

votes

**2**answers

248 views

### Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$

I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...

**4**

votes

**1**answer

157 views

### Discrete spectrum and almost periodicity

According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to ...

**5**

votes

**1**answer

105 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 ...

**11**

votes

**2**answers

462 views

### Blinking graphs

For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a bit assignment for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the ...

**0**

votes

**0**answers

47 views

### Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space.
I am interested in informative examples and applications of such systems. I know ...

**0**

votes

**0**answers

73 views

### Is the Rossler attractor globally stable?

The standard form of the Rossler system is $\frac{dx}{dt}=-y-z$, $\frac{dy}{dt}=-x+ay$, $\frac{dz}{dt}=b+z(x-c)$. For simplicity, consider the well known chaotic attractor that exists at a=b=0.2, ...

**4**

votes

**1**answer

198 views

### What are the generating partitions of the odometer?

According to the countable generator theorem, every ergodic invertible measure-preserving transformation has a generating partition.
What are the generating partitions of the dyadic odometer ? I ...

**0**

votes

**0**answers

40 views

### Transformations whose product with the odometer are ergodic

Let $T$ be an invertible ergodic transformation on a Lebesgue space $X$ and $O$ be the dyadic odometer on $(0,1)$. Is it true that $T\times O$ is ergodic if and only if $T^{2^n}$ is ergodic for every ...

**2**

votes

**0**answers

53 views

### Techniques for the analysis of interacting particle systems with a finite number of particles, which do not resort to limiting arguments?

I am interested in pointers to (keywords/authors) recent research on the analysis of interacting particle systems with a finite number of particles which do not resort to limiting arguments converting ...

**2**

votes

**2**answers

92 views

### Transformations whose product with a given rotation are ergodic

I am interested in the ergodic (invertible) transformations $T$ such that $T\times R_\theta$ is ergodic where $R_\theta$ is the rotation on $S^1$ with a given irrational angle $\theta$ (not all ...

**4**

votes

**0**answers

149 views

### Dynamical Mordell-Lang on Kahler manifolds?

Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...

**3**

votes

**0**answers

82 views

### Reconstructing a vector field on the circle

Consider a ODE on the circle of the form
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \omega(x) \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix} \begin{pmatrix} x_1 \\ ...

**6**

votes

**1**answer

90 views

### Sign problem in a Calogero-Moser system: proof of integrability?

Everyone of us had sometimes this awful feeling that some sign is lost in a calculation and that this sign is perturbing some fundamental understanding of what is going on. I feel the same has ...

**3**

votes

**1**answer

146 views

### Transformation extending all ergodic rotations

Is there an invertible measure-preserving transformation (preferably a nice one) admitting every irrational rotation as a factor ? I guess the spectrum is the relevant tool to address this question ...

**1**

vote

**1**answer

257 views

### Given $g(h(z))$ is convergent, what can be said about the convergence of $g(z)$ and $h(z)$? [closed]

Consider the iterated function $f^t(z)=f(f(f(...f(z))...))$ where $t \in \mathbb Z$ and $f(z)$ is convergent. Then the iterates of $f(z)$ such as $f^2(z), f^3(z), f^4(z)$ are convergent. Now let $r ...

**1**

vote

**2**answers

111 views

### What are good references for spatial dynamics?

Hello I have started working on my PhD a short while ago and wondered if there might be any good introductions to spatial dynamics.
I have a basic understanding of dynamical systems but would like to ...

**0**

votes

**2**answers

183 views

### If Non wandering Set is whole space then recurrent set is dense?

I tried to prove following statement and use some techniques but I couldn't get result :
Question: If Non wandering Set is whole space then Recurent set is dense??
when $T:X \to X$ is ...

**0**

votes

**0**answers

78 views

### The growth rate of almost periods for almost periodic function

A subset $A \subset \mathbb{R}^2$ is relative dense if there exists $L>0$ such that for every $p\in \mathbb{R}^2$ there exists $p' \in A$ such that $|p-p'|<L.$
A continuous function $f : ...

**0**

votes

**0**answers

63 views

### Joint point of coarse geometry and dynamical system?

My major interest is on dynamical systems,
but I did REU in a coarse embedding problem.
I wonder whether there's some significant connection between those two subjects.
I've tried to google for a ...

**5**

votes

**0**answers

102 views

### Topologically transitive, pointwise minimal systems

I'm cross-posting this from SE.
Let $T$ be a group, and let $(X,T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically ...

**4**

votes

**1**answer

131 views

### Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation.
Consider a measurable set $A\subset X$ with $0<\mu(A)<1$
For each $N$ define ...

**1**

vote

**1**answer

221 views

### A quantitative Kronecker theorem

I encounter the following question.
$\textbf{Problem}$: For almost all Matrix $M\in\mathcal M_{m\times n}(\mathbb R),$ all $y\in \mathbb R^m$ and any $N$, small $\epsilon>0$, there exists a ...

**1**

vote

**0**answers

93 views

### Strict factor of a dynamical system with the same entropy [closed]

Say that a factor of an invertible measure-preserving transformation $T$ is strict if it is not isomorphic to $T$. Does there exist an invertibe mpt $T$ such that $0 < h(T) < \infty$ and ...

**7**

votes

**0**answers

250 views

### Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...

**8**

votes

**1**answer

143 views

### How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
...

**6**

votes

**1**answer

121 views

### Generator determined by finitely many translates implies zero entropy

Let $T$ be a measure preserving transformation of a standard probability space $(X,\mathcal{B},\mu)$. A partition $\alpha$ of $X$ is said to be a generator for $T$ if the smallest $T$ invariant ...

**7**

votes

**2**answers

308 views

### Periodicity in iterated powers of sin, cos, exp

Given a complex number $z$, consider the sequence
\begin{align*}
a_0 & = 1\\
a_1 & = (cos(1))^z\\
a_n & = (cos(a_{n-1}))^z
\end{align*}
This question is about trying to understand ...

**0**

votes

**1**answer

56 views

### Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$

This question has been asked here but there is no answer:
http://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
Consider autonomous ODE $y' = ...