# Tagged Questions

**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

139 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

287 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

438 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

202 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

236 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

198 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

111 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

100 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

165 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. ...

**3**

votes

**0**answers

71 views

### Question about a length inequality in algebraic dynamics

Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of ...

**2**

votes

**1**answer

223 views

### Open dynamical system of doubling map

I want to prove the following simple lemma:
Let $T(x)=2x\mod 1$, be defined on $[0\,,1)$ to itself,suppose for any $k\ge 0$, $T^{k}(a)\notin(a\,,b)$, then we have
$\Omega=\{x\in[0\,,1):\mbox{for ...

**5**

votes

**0**answers

574 views

### A question on “The weakened Hilbert 16th problem”

In this question we are interested in the number of limit cycles which appear in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
...

**3**

votes

**1**answer

169 views

### Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$?
(Possibly by perturbing a rotation in the real-analytic topology?)

**2**

votes

**0**answers

93 views

### Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?

I would be interested to know if a certain set of periodic solutions for
the two-dimensional Navier-Stokes equations is closed generically.
Many similar (yet not identical) set-ups can be found in the ...

**2**

votes

**2**answers

204 views

### Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...

**2**

votes

**1**answer

109 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

99 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

282 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

106 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

117 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

150 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

269 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

132 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

158 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

44 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

354 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

401 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 ...

**71**

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

121 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

86 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

296 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

210 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

115 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), ...

**3**

votes

**1**answer

175 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

136 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

86 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

99 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

823 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

137 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

119 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

219 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

173 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

134 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

115 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

179 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

167 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 ...