# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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### Baker map-like problem

Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...

**2**

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66 views

### How to compute expansion factors for hyperbolic rational maps?

It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...

**7**

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**2**answers

328 views

### Is it possible to prove unboundedness of 3rd order ODE?

Consider the 3rd order ODE
$$\dddot{x}+A\ddot{x}-\dot{x}^{2}+x=0$$ where $\dot{x}\equiv \frac{dx}{dt},\ddot{x}\equiv \frac{d^{2}x}{dt^{2}}, etc$. $A$ is a constant.
If we multiply this equation by $\...

**2**

votes

**1**answer

94 views

### Irreducible subcontinuum of Lorenz attractor?

In my first question Lorenz attractor path-connected?, some are saying the Lorenz attractor $\mathscr L$ is not path-connected.
But suppose $x$ and $y$ are two points in different path components of ...

**9**

votes

**1**answer

561 views

### Lorenz attractor path-connected?

Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.
EDIT: The answer below is unsatisfactory, and possibly ...

**7**

votes

**1**answer

104 views

### Under which conditions do ellipsoids have a focal property?

Given an ellipsoid $E$, we consider the trajectories of light inside $E$ assuming that $\partial E$ would be a mirror. In other words, let a light trajectory be piecewise linear path $\gamma:[0,\infty)...

**1**

vote

**1**answer

102 views

### Increasing union of embedded submanifold is immersed manifold

While working on the proof of the stable manifold theorem, I came across a problem that I'm not able to really grasp. Given some Anosov map $f: M \to M$ on a compact Riemann manifold $M$, one can ...

**5**

votes

**1**answer

160 views

### Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$

If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...

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**2**answers

93 views

### Quantitative bound on irrational rotation recurrence time

Given an irrational $a$, the sequence $b_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified.
Given a point $p$ in $\mathbb ...

**1**

vote

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69 views

### $C^1$-foliation are absolutely continuous

Brin & Stuck defined in Introduction to dynamical system two notions:
That of a absolutely continuous foliation : given any foliated chart $U$ on some Riemannian manifold $M$ (with foliation $W$),...

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210 views

### Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...

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22 views

### Splittings for generic flows on a vector bundle

Question: Consider a smooth vector bundle $\pi:V\to B$ and the space $\mathcal{F}$ of $C^k$ linear flows $\Bbb{R}\times V \to V$ endowed with the strong $C^k$ Whitney topology. Is it true that for all ...

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27 views

### Rotation rates for a linear flow on a vector bundle

The following linear ODE on $\Bbb{C}$
$\dot{z} = (a + i b)z$
has solutions $z(t) = e^{(a+ib)t} z(0)$. Hence the real part $a$ captures expansion rate and the imaginary part $b$ captures rotation ...

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74 views

### Gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded.
To be more precise, a ...

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**2**answers

55 views

### Lower bound of positive entropies of automorphisms on tori

Let $A$ be an automorphism on tori $\mathbb{T}^d$. It is well known that the topological entropy
$$
h(A)=\sum_{\lambda} \max\{0, \log|\lambda| \}
$$
where $\lambda$ goes through all eigenvalue of $A$ ...

**3**

votes

**1**answer

89 views

### On Krieger's Embedding Theorem

This is Theorem 10.1.1 of Lind & Marcus's book, An Introduction to Symbolic Dynamics and Coding. They say that is "straightfordward" to go from
Let $X$ a shift of finite type and $Y$ a mixing ...

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58 views

### What's equal the below power nested radical?

The same copy of this question is montioned here in SE with no convinced Answer , I want to know what MO will say about the below nested radical as a power form
it is well known that
$$\frac{2}{\...

**3**

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**0**answers

61 views

### Flow of zeros in the shifted exponential generating function?

Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...

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58 views

### Time-varying perturbations of continuous-time hyperbolic orbits

My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then ...

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85 views

### Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently:
Let $X$ be a compact $k$-...

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**0**answers

89 views

### Bifurcations due to a nonlinearity parameter

Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...

**9**

votes

**1**answer

178 views

### Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...

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86 views

### stochastical stable

Given dynamic $f: S^1 \to S^1$ with Lebegue measure $dm$ on $S^1$. Assume it has unique SRB probability measure $\frac{d\mu_f}{dm} dm $. Given left shift space $([-\epsilon, \epsilon]^{\otimes \...

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175 views

### Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...

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votes

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111 views

### A generalized Furstenberg's $\times p,\times q$-conjecture

Let $p,q$ be two positive integers such that $\frac{\log p}{\log q}\notin\mathbb{Q}$. Furstenberg's $\times p,\times q$ conjecture says that the only ergodic nonatomic $\times p,\times q$-invariant ...

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55 views

### Relations between Omega, Local Indicable and Right Orderable groups

We know that the set of Right-Orderable groups $RO$, is contained in the set of $\Omega$- groups (Read it from "A Note on Group Rings of Certain Torsion-Free Groups" Burns-Hale).
A Group $G$ is a ...

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vote

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42 views

### Solutions of nonlinear equations with multiple parameters

In the course of analysing a particular three dimensional nonlinear dynamical system, I find the need to solve a nonlinear equation of the form:
$$ \mathcal{M}(x, \lambda) := x - f(x, \lambda_1, \...

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votes

**1**answer

187 views

### Distribution of $\{cn^a\}$

Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) ...

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**1**answer

324 views

### Do solenoids embed into Möbius strips?

I found a strange attractor which looks a lot like a solenoid. The attractor continuum is the closure of a continuous line which limits onto itself, and it is locally homeomorphic to Cantor set times ...

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votes

**2**answers

190 views

### Free ergodic probability measure-preserving actions of the free group

Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group.
An action of $\Gamma$ on $X$ is:
essentially free if for all $g \in \Gamma \setminus \{e \}$,...

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vote

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84 views

### size of local strong stable manifold is measurable

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...

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votes

**1**answer

477 views

### Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?

Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...

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79 views

### Is the Mandelbrot set weakly self-similar?

A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...

**1**

vote

**1**answer

63 views

### Lotka Volterra existence of Caratheodory solution

I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$
has a unique Caratheodory solution ...

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192 views

### About the limit of transverse intersection

Let $n$ be a fixed positive integer, and let $W^{s}(R_{q})$ and $W^{u}(R_{q})$ be the stable and unstable manifolds of a fixed point $R_{q}$ of a discrete 2-D mapping $f_{q}$. Notice that the sequence ...

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votes

**2**answers

232 views

### Newton method and Siegel disks

I am looking for a degree 3 polynomial $P$ whose associated Newton's method $z \mapsto z - P(z)/P'(z)$ has a Siegel disk. Is there an explicit example of such polynomial $P$?

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votes

**3**answers

206 views

### Random reflections unexpectedly produce banded distributions

Start with $p_1$ a random point on the origin-centered unit circle $C$.
At step $i$, select a random point $q_i$ on $C$, and a random mirror line
$M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...

**4**

votes

**1**answer

175 views

### Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...

**2**

votes

**1**answer

145 views

### Irreducible but not completely irreducible

Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <\infty$).
Let $F:M\times \mathbb R^d\to M\times \mathbb R^d$ be a linear cocycle, ...

**2**

votes

**1**answer

184 views

### continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.
Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...

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vote

**1**answer

67 views

### Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system:
\begin{equation}
\mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1]
\end{equation}
I would adapt the definition of marginal stability from this ...

**6**

votes

**0**answers

73 views

### Stable commutator lengths of pseudo-Anosovs

Does anyone have an example of a pseudo-Anosov mapping class for which the stable commutator length is known exactly?

**2**

votes

**1**answer

87 views

### stationary measure for linear cocycle(random transformation matrices)

Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...

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124 views

### How was the pair of pants introduced [closed]

There are many results mentioned pairs of pants, and it seems to be a classical model. Why are the pairs of pants so useful?
For example, does it have any application if we estimate the perimeter or ...

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votes

**2**answers

331 views

### Minimal, uniquely ergodic but not Lebesgue-ergodic?

So here's my question:
Does there exist a minimal diffeomorphism of class at least $\mathcal{C^2}$ of a compact manifold X which is
minimal
uniquely ergodic with unique probability measure $\mu$
...

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votes

**2**answers

414 views

### Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice.
I have heard that, for any real number $\...

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vote

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30 views

### definition of mixing component

definition of ergodic component: consider stationary dynamical system $(X, \mathcal{B}, \mu, T)$, each ergodic component is $m(\cdot)=\mathbb{E}_{\mu}^{\mathcal{I}}\mathbf{1}_{(\cdot)}$, $\mathcal{I} $...

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90 views

### Is $\partial M_d$ continuously determined by $d$?

This question is inspired by a question on math.stackexchange:
https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089
The animation ...

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**0**answers

94 views

### How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...

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votes

**1**answer

64 views

### Does differentiating an integro-differential equation results in equivalent stability of the solution?

I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation:
...