**2**

votes

**1**answer

111 views

### Power series expansion of the Koenigs function

Given a non-zero holomorphic function $f$ fixing $0$ which isn't a Mobius transform, the Koenigs function of $f$, which we'll call $h$, is the function which linearizes $f$ in the sense that
$$
...

**2**

votes

**0**answers

112 views

### Existence and uniqueness of heteroclinic orbits

I am looking for conditions on a nonlinear dynamical system $\frac{d\vec{x}}{dt} = \vec{F}(\vec{x})$ that guarantee the existence of a unique heteroclinic orbit between a stable attractor of this ...

**7**

votes

**2**answers

285 views

### Iteration of a 2D map involving absolute value: phase transition?

I was looking at this map: $f(x,y) \mapsto (|x-y|,x)$, starting from some point
with coordinates $(x,y) \in [0,1]^2$, and iterating:
$(x,y),\, f(x,y), \, f^2(x,y), \,f^3(x,y), \ldots$.
It displays ...

**3**

votes

**1**answer

110 views

### How is the Fractal Dimension of a Parametric Curve Related to the Fractal Dimensions of its Coordinate Functions?

Question:
Suppose the fractal dimension $1\le d_c\le2$ of a planar parametric curve $c(t) := (x(t),y(t))$ is given;
can any nontrivial estimates for the fractal dimensions $d_x$ of $x(t)$ and $d_y$ of ...

**2**

votes

**1**answer

119 views

### Does conjugacy preserve the set of synchronizing blocks?

A synchronized system is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then ...

**2**

votes

**2**answers

154 views

### Mixing coded systems and period of their graph presentations

A coded system [see F. Blanchard, G. Hansel, Systèmes codés, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8.
...

**5**

votes

**2**answers

278 views

### Lebesgue entropy zero and positive topological entropy

I am looking for examples of volume preserving $C^{\infty}$ diffeomorphisms $f$ of a surface, which have positive topological entropy ($h(f) > 0$), but that the Lebesgue measure entropy (metric ...

**3**

votes

**1**answer

135 views

### Automorphisms of strictly ergodic shift spaces

Let $X$ be a strictly ergodic shift space, and $\omega_1$, $\omega_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega_1)=\omega_2$? By an automorphism I ...

**1**

vote

**1**answer

160 views

### There is a horseshoe with positive measure

Here is a theorem by Bowen :
My question is about the highlighted part in the picture. why there such a function $g$ exist?

**1**

vote

**0**answers

45 views

### Id monodromy in hamiltonian dynamics

In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t-1)q^2$, $a,b$ - parameters). From numerical ...

**12**

votes

**2**answers

382 views

### Are rounded rectangle billiard dynamics ergodic?

Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.
(Image from this link.)
Q. Is it known that the ...

**13**

votes

**0**answers

408 views

### Blocking light with mirrored convex objects

There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...

**72**

votes

**2**answers

106k views

### Perfectly centered break of a perfectly aligned pool ball rack

Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a ...

**0**

votes

**1**answer

124 views

### An absolutely continuous foliation, which is not transversely absolutely continuous

Currently I'm studying "Introduction to dyamical systems" by Stuck and Brin. In chapter 6 they define absolutely continuous and transversely absolutely continuous foliations. By proposition 6.2.2 if ...

**2**

votes

**0**answers

89 views

### random maass waveforms

Let $H$ be the upper half complex plane and $\Gamma$ a discrete subgroup of $SL_2(\mathbb{Z})$ such that the volume of of $\Gamma \backslash H$ is finite. There is a conjecture of Berry that Maass ...

**0**

votes

**1**answer

88 views

### Looking for methods/results for explicitly bounding iterations of rational functions

In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series.
That is, suppose that
$$
...

**9**

votes

**1**answer

304 views

### An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part.
1.Is there a polynomial Hamiltonian ...

**2**

votes

**1**answer

214 views

### Angle between two subspaces

Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space
$T_xM=E^s(x)\oplus ...

**1**

vote

**0**answers

193 views

### Question on measure zero set of initial conditions in dynamical systems

[Update] Let $S \subseteq \mathbb{R}^n$ be a closed, bounded, convex set with measure $m(S)>0$ and let an autonomous dynamical system (system of ODEs) be given by
$$\frac{dx}{dt} = f(x),$$
where ...

**2**

votes

**1**answer

119 views

### the union of local stable manifolds along local unstable manifolds

Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), ...

**4**

votes

**1**answer

180 views

### A question about ergodicity

Let $X$ be a compact metric space, $T:X\rightarrow X$ a homeomorphism and $\mu$ be a $T$-invariant probability measure on $X$ such that the set of points with dense orbit in $supp(\mu)$ has full ...

**3**

votes

**2**answers

139 views

### Decay of Correlation, references for a non-standard way

In ergodic theory is common to use the decay of correlation property to deduce
properties analogues to those of i.i.d. random variables.
Call $X\doteq [0,1].$
Examples of decay of correlation ...

**2**

votes

**0**answers

90 views

### uniquely ergodic hyperbolic invariant set

The question is to classify uniformly hyperbolic invariant sets supporting uniquely ergodic invariant measure. The only examples that I expect are: fixed points, periodic orbits and Cantori(Denjoy ...

**4**

votes

**1**answer

100 views

### For a linear dynamic system, what can we learn from its singluar value and rank?

Given a linear system $\frac{dx}{dt}=Mx$, what's the relationship between the dynamic's property and the singular value decomposition/rank of $M$ ?

**19**

votes

**5**answers

827 views

### Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...

**0**

votes

**0**answers

138 views

### Express measurable entropy in terms of Fourier coefficients of the measure

Let $S^1$ be the unit circle , $\mu$ be a Borel probability measure on $S^1$ and
$T:S^1\to S^1$ is a measure-preserving map (not necessarily invertible) with respect to $\mu$. The Fourier ...

**2**

votes

**1**answer

123 views

### expansive continuous flow

I encounter with two definitions for expansive continuous flows and their equivalence is unclear for me. Could anyone can explain for me please? Thanks in advance. I cite below these two definitions.
...

**3**

votes

**1**answer

226 views

### Julia sets without Montel's theorem

Let $J(c)$ be the Julia set of $f(z)=z^2 +c$ defined as the closure of repelling periodic orbits. Is there a way to prove that $J(c)$ is the boundary of the basin of attraction of attractive fix ...

**5**

votes

**1**answer

180 views

### Topological classification of Morse-Smale flows

Does anyone know of papers that mention the classification of non-singular Morse-Smale (NMS) flows up to topological equivalency? I am particularly interested in the flows on manifolds of dimension 3. ...

**5**

votes

**1**answer

140 views

### Is there a similar theorem in the partially hyperbolic case?

Theorem 5.10.3 from Introduction to dynamical systems, by Brin & Stuck:
Let $f:M\rightarrow M$ be an Anosov diffeomorphism. Then the following are equivalent:
$NW(f)=M$,
every unstable manifold ...

**3**

votes

**1**answer

118 views

### Convergence of trajectories and asymptotic stability

Say that an autonomous system $\dot{u} = f(u)$ in $\mathbb{R}^{m}$ has the property that for any two solutions $x(t), y(t)$ corresponding to initial conditions $x(0)$ and $y(0)$ the trajectories are ...

**1**

vote

**0**answers

181 views

### Examples of amenable groups acting on the real line

In the literature, there are several examples of solvable groups acting by order preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth acting in the same ...

**7**

votes

**2**answers

171 views

### Well-definedness of single-particle smooth billiards flow

Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well ...

**6**

votes

**1**answer

190 views

### Approximating an iteratively defined function

Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows:
$$f_0(x) =1+2x$$
$$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } ...

**3**

votes

**2**answers

231 views

### Systems similar to Erdős numbers?

As many mathematicians know, each person has an Erdős number (see: http://en.wikipedia.org/wiki/Erd%C5%91s_number). That is, Erdős himself has Erdős number zero, each person who published anything ...

**2**

votes

**1**answer

156 views

### The centralizer of Lienard equation

Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...

**15**

votes

**1**answer

760 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

**4**

votes

**1**answer

203 views

### Volume-preserving mappings in the torus $T^n$

Let $T^n$ be the $n$-dimensional torus and let $F$ be the set of all volume preserving continuous mappings $f:T^n\to T^n$. I would like to know if $F$ is connected in the sense that for any $f\in F$ ...

**8**

votes

**2**answers

635 views

### Getting unique ergodicity from minimality

It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask whether the implication holds in following particular situation:
Suppose $X$ is a compact space, $f:X \to ...

**3**

votes

**0**answers

106 views

### Pointwise ergodic theorem for amenable semigroups

Using tempered Folner sequences one may show a pointwise ergodic theorem for amenable groups.
(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)
Is there a similar ...

**3**

votes

**1**answer

229 views

### Uniform convergence of Birkhoff averages and unique ergodicity

I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET):
Let $T$ be a ...

**0**

votes

**0**answers

137 views

### Prove that origin is globally exponentially stable with Lyapunov Indirect Method

I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially ...

**1**

vote

**1**answer

174 views

### Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...

**2**

votes

**1**answer

302 views

### Dynamics of Master Equation

I'm going to do research on dynamics of master equation of $n$ states
$$\dot p_i=A_{ij}p_j\qquad i=1\ldots n$$
where $p_i$ is the $i$-th component of probability vector and $A_{ij}$ is transition rate ...

**2**

votes

**0**answers

86 views

### Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action

My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...

**9**

votes

**0**answers

230 views

### Periodic orbits of a spinning ball in a square

Periodic orbits of a billiard ball bouncing in a square have been well-studied.
I am seeking similar analysis of what is sometimes called a rough ball, one
whose high friction causes it to pick up ...

**3**

votes

**1**answer

130 views

### Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s?

A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by ...

**9**

votes

**1**answer

272 views

### Random circle rotations

Weyl's equidistribution theorem states that the orbit of a point on the circle under rotation by $\alpha$ becomes asymptotically equidistributed with respect to Lebesgue (Haar) measure whenever ...

**11**

votes

**3**answers

560 views

### How to draw a Zoll surface?

I take into account that lots of questions on Zoll surfaces have already been asked on the forum. But I will stubbornly continue asking. Are there any chances to draw explicitely at least one Zoll ...

**3**

votes

**1**answer

50 views

### Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3
$$x'(t)=f(x(t),d)$$
where the dynamics f is homogeneous of degree 1 and there is exactly one
line of equilibrium points. This line is independent of the ...