**4**

votes

**3**answers

314 views

### Precise location of the Mandelbrot Bulb Attachment to the main Cardioid

Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...

**0**

votes

**1**answer

152 views

### Second order ODE

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?
$$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$
$C$ ...

**1**

vote

**2**answers

369 views

### $\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$

Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any ...

**5**

votes

**1**answer

116 views

### Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function,
...

**2**

votes

**0**answers

77 views

### Automorphisms of Nilmanifolds

Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...

**1**

vote

**1**answer

69 views

### A question for the inverse orbit in the construction of conformal measure

Recently, I read a theorem of existence of conformal measure for the rational map.
I did not understand two places in the proof. The author claims that
there exists an open set $V\subset ...

**11**

votes

**2**answers

462 views

### Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...

**1**

vote

**1**answer

176 views

### Fixed point of a function on the circle

Consider a circle $C$ with radius of $r$, we place $m$ balls(treated as point) randomly on it, and each ball $i$ has the mass $m_i$. We define a function $\varphi:C\rightarrow C$ which maps $x\in C$ ...

**0**

votes

**0**answers

115 views

### Manifolds supporting finite order diffeomorphisms (a local construction?)

The following question is mainly inspired by this previous one Which manifolds admit a diffeomorphism of order $n$? and some answers given there.
For $d\geq 2$, let $\mathbb{B}^d$ denote the closed ...

**2**

votes

**2**answers

208 views

### Classes of dynamical systems

A consequence of Birkhoff ergodic theorem tells us that ergodicity is equivalent to:
$\forall A,B \in \mathcal{B} \quad \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to ...

**1**

vote

**1**answer

75 views

### Generators for the affine automorphism group of the octagon

Consider an octagon $O$ with opposite edges identified. Lemma 3.2.4 of (1) (subscription link) claims that the affine automorphism group of $O$ is generated by $D_8$ and the shear $\sigma$ such that, ...

**5**

votes

**1**answer

91 views

### Unfoldings of trajectories on the Veech triangle $V_4$

Let $V_4$ be the isosceles triangle with base angle $\pi/8$. $V_4$ is a Veech triangle, so the dynamics of billiards on it are very well understood.
Above is the unfolding of $V_4$, with edge ...

**4**

votes

**0**answers

144 views

### introduction books for Dynamic systems of discrete Schrodinger operator for beginner

In this semester, I study in a class of dynamic system. recently the French professor turn to the dynamic system of discrete operator. I find it is difficult to find a book in English. (I have found ...

**2**

votes

**1**answer

130 views

### $C^1$ stability conjecture on non-compact manifolds

In the study of dynamical systems, the link between structural stability and Smale's Axiom A has been explored by many authors. One of the important outcomes of this endeavor was the proof of $C^1$ ...

**6**

votes

**3**answers

208 views

### Existence of nonergodic polygonal billiard

Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one.
A standard conjecture is that a ...

**2**

votes

**1**answer

101 views

### Centre manifold theory for a curve of equilibrium points

I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001).
The part which is known
Let's consider a differential equation in higher-dimensional ...

**2**

votes

**1**answer

157 views

### Non-hyperbolic fixed points in multidimensional systems

Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start ...

**2**

votes

**1**answer

74 views

### Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end.
$\mathrm{\underline{Background}}$
Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...

**5**

votes

**0**answers

231 views

### Deterministic shifts

We consider (topological) dynamical systems $(\Omega, S)$, where $S$ is the shift $(Sx)_n=x_{n+1}$, and $\Omega\subset[0,1]^{\mathbb Z}$ is a compact, shift invariant subspace. I call such a system ...

**4**

votes

**1**answer

140 views

### Graph presentation of Lexicographic shifts

Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma_n = \mathop{\prod}\limits_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic ...

**3**

votes

**2**answers

182 views

### Nielsen-Thurston classification of homeomorphisms for open surfaces?

In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...

**4**

votes

**1**answer

160 views

### Homeomorphisms that admit a decomposition

Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$.
If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ...

**2**

votes

**0**answers

83 views

### Quick estimate of attractor of non-linear dynamical system [closed]

Say I have a system of form
$$
\frac{dy}{dt} = f(y),
$$
and it is know this system has an attractor. Can I quickly for given $\varepsilon$ guess some point, such in its $\varepsilon$- neighbourhood ...

**2**

votes

**1**answer

114 views

### Omit each vertex in turn of convex polygon: Iterative limit?

Let $P=P_0$ be a convex polygon of $n$ vertices $v_k$.
Let $P_{i+1}$ be the convex polygon obtained by intersecting the halfplanes
determined by the lines through every other vertex.
Below, $P_0$ is ...

**7**

votes

**6**answers

623 views

### Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a ...

**4**

votes

**1**answer

112 views

### Automorphism group of compact abelian group

I am looking for references on the automorphism group $\mathrm{Aut}(X)$ of a compact abelian group $X$. By automorphisms I mean topological group automorphisms. Some particular questions are as ...

**2**

votes

**0**answers

78 views

### Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic

Consider mechanical Hamiltonian system of the form
$$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$
Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian ...

**4**

votes

**0**answers

94 views

### Example for a dynamical system which is not point-distal

Let $(X, d)$ be a compact metric space, let $T$ be a group of actions on $X$. Then $(X,T)$ is a topological dynamical system with transformation group $T$, and we denote it by $(X,T)$. We say points ...

**1**

vote

**0**answers

92 views

### Periodic solution of first order ODE

There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system
$$
\left\{
\begin{array}{l}
\dot{x}=f(x), \\
x(t_0)=x_0,
...

**3**

votes

**0**answers

115 views

### Nonexistence of Limit Cycle

Consider a planar dynamical system described in polar coordinates as
$$
\left\{
\begin{array}{ll}
\dot{\theta}=\Delta - r \sin \theta,\\
\dot{r} = - r + 1 + \cos \theta,
\end{array}
\right.
$$
where ...

**3**

votes

**2**answers

152 views

### Mixing property of first return map

Let $(X,\mathcal{X},\mu)$ be a probability measure system, $T:X\to X$ be a $\mu$-preserving isomorphism on $X$. Let $A\in \mathcal{X}$ such that $\mu(\bigcup_{n\ge 0}T^nA)=1$, and $\mu_A$ be the ...

**4**

votes

**2**answers

491 views

### Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...

**1**

vote

**1**answer

172 views

### Analytic vector fields on surfaces which have infinite number of singularities

Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$.
A local question
Is there an analytic vector ...

**2**

votes

**1**answer

128 views

### Global Solutions of Ordinary Differential Equations

Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...

**7**

votes

**1**answer

368 views

### Raphael Douady's thesis: Applications du théorème des tores invariants

Raphael Douady's thesis, Applications du théorème des tores invariants, has been cited in numerous papers by many experts.
According Wikipedia, he proves of the equivalence of KAM ...

**2**

votes

**1**answer

81 views

### Constructing an interval exchange given a prescribed trajectory

Given a prescribed trajectory, is it possible to construct an interval exchange having this trajectory?
For example, given a 3-letter word (like aaabbbccabcaaa ), is it possible to construct a 3- ...

**14**

votes

**2**answers

472 views

### Spearing rolling hula hoops

Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, ...

**2**

votes

**0**answers

70 views

### Link between presence of attracting random fixed points and synchronisation - is this an open question?

This is a question in the theory of random dynamical systems.
Let $(X,d)$ be a compact metric space, let $(I,\mathcal{I},\nu)$ be a probability space, and let $(f_\alpha)_{\alpha \in I}$ be an ...

**0**

votes

**0**answers

47 views

### Question on center-stable manifold

Assume that you have a gradient system smooth enough and a fixed point $x_{0}$. Is it true that if $x_{0} \in \omega(x)$ then $\gamma^{+}(x)$ must intersect the local center-stable manifold of ...

**25**

votes

**2**answers

1k views

### Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ ...

**0**

votes

**1**answer

134 views

### Find a sequence with uniform frequencies and recurrent property

Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$
such that this sequence has uniform ...

**0**

votes

**0**answers

26 views

### n-body systems and bifurcations

From what I understand, in bifurcation theory, one definition of the equivalence of two dynamical systems is that they are topologically equivalent. However if say proteins A, B start out as straight ...

**3**

votes

**2**answers

252 views

### Physical Measure Vs. SRB measures

Anybody can help me to have an idea about an example showing the difference of a Physical measur with compare to an SRB measure?
By a Physical measure i mean in the sense of $\nu$ a ...

**11**

votes

**2**answers

437 views

### Invariant subsets of $z \mapsto z^2$

Where can I find an explicit construction of closed invariant subsets of the map $z \mapsto z^2$ on the unit circle? Furstenberg mentions that there are continuum of such disjoint minimal sets but ...

**4**

votes

**2**answers

200 views

### Invariant measures on a compact metric space

I'm dealing with a continuous flow on a compact metric space $X$, and $\mu$, $\nu$ are two invariant Borel probability measures on $X$. If I know that $\mu(A)=\nu(A)$ for all the invariant Borel ...

**0**

votes

**1**answer

230 views

### A variation of the Banach fixed-point theorem

Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem ...

**6**

votes

**1**answer

196 views

### Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture

Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...

**1**

vote

**0**answers

108 views

### Gradient-like systems and limit sets

Suppose we have an autonomous ode $$\dot{x} = f(x)$$ which corresponds to a $\underline{gradient-like}$ dynamical system, $f: \mathbb{R}^n \to \mathbb{R}^n$. Is it true that the set of initial ...

**0**

votes

**0**answers

99 views

### excplicit formula of iterates of an interval exchange

Let $f$ be an interval exchange transformation of $[0,1]$. Is there an explicit formula giving $f^k(0)$ in function of $k$?
If not, are there particular cases where this formula is simple? (except ...

**3**

votes

**2**answers

163 views

### Markov Partitions for toral automorphisms

I know that my question is more practical than theoretical. But, I do know where to look for the theoretical sources.
I want to find a program in the case that it exists (does it?), or to program it. ...