**1**

vote

**0**answers

171 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

160 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

179 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

219 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

149 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

704 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

187 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

359 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 ...

**2**

votes

**0**answers

98 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 ...

**2**

votes

**1**answer

192 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

114 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

171 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

300 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

83 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

215 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 ...

**2**

votes

**1**answer

110 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

244 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 ...

**9**

votes

**2**answers

399 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

45 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 ...

**2**

votes

**3**answers

174 views

### Non-linear state-space model system stability using Lyapunov?

I have a non-linear system modelled in state-space as follow:
$$
\mathbf {\dot x} = \mathbf A(x) \mathbf x
$$
I need to find out if this system is stable, so I was thinking in using the Lyapunov ...

**1**

vote

**0**answers

79 views

### A argument related measurable partitions in dynamic system

$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$ X_{0}$ the set of all points $x\in X$, such that for all sufficiently small ...

**0**

votes

**3**answers

325 views

### Good books on Geometric Theory of Dynamical Systems

I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...

**1**

vote

**1**answer

90 views

### Stability of a system of ODEs

It is well known that for a system of ODEs, $\dot{\boldsymbol{y}} = \boldsymbol{Ay}$, the global stable equlibrium point is given by the eigenvector that correponds to the largest eigenvalue of ...

**3**

votes

**0**answers

184 views

### The $\Omega$-Stability Theorem

I'm currently studying the $\Omega$-Stability Theorem:
Theorem: If $\mathcal{R}(f)$ has a hyperbolic structure then $f$ is $\mathcal{R}$-stable.
Some explanations about the statement: $f$ is a $C^1$ ...

**15**

votes

**2**answers

477 views

### “Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence.
Start with a polygon, and replace it by locating a point on every edge
a fraction $\alpha$ between the two endpoints. For ...

**1**

vote

**0**answers

41 views

### Discrete Optimal Control and Monotone Policies

Let $x = (x_1,x_2) \in \mathbb{N}^2$ be the state, $u$ be the control, and the dynamics be given by $x^{(k+1)} = f(x^{(k)}, u^{(k)}, w^{(k)})$ where $w^{(k)}$ is an IID noise source. For some stage ...

**9**

votes

**1**answer

350 views

### Do quantum “Sure-Shor separators” have a natural Veronese/Segre classification? (question inspired by Gil Kalai and Aram Harrow)

Aram Harrow asked: "Is there any place this is written up?"
Update Partly in answer to Aram's question, the thermodynamical properties of varietal dynamical systems now are written-up in ...

**5**

votes

**0**answers

105 views

### Using topological pressure to determine a subshift of finite type

I am interested in recognising graphs (or matrices, or subshifts of finite type) using topological pressure. Suppose that we play the following game:
${\bf Step 1:}$ I write down an irreducible n x n ...

**1**

vote

**0**answers

73 views

### Structural stability of the Chen system

Structural stability of the Chen system
The concept of structural stability was introduced by Andronov and Pontryagin in 1937, and it has a crucial role in dynamical systems theory. Conditions for ...

**5**

votes

**1**answer

130 views

### Entire functions with a null real escaping set

Let $f$ be a entire function (stable on $\mathbb{R}$), and $E_{\mathbb{R}}$ its real escaping set : $$E_{\mathbb{R}} = \{ x \in \mathbb{R} : f^{(k)}(x) \rightarrow_{k \to \infty} \infty \} $$
We put ...

**4**

votes

**1**answer

171 views

### Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set :
$$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$
Question : If $K(f)$ ...

**4**

votes

**1**answer

204 views

### Katok's conjecture on entropy and periodic orbits for generic $C^1$ diffeomorphisms

Let $M$ be a compact finite-dimensional manifold and $f\colon M\to M$ be a diffeomrphism. By $P_n(f)$ we denote the number of periodic points of $f$ with period $n$, that is, the number of fixed ...

**1**

vote

**0**answers

53 views

### On global attraction of a stable node for a four dimensional nonlinear system

Consider the dynamical system on ${\mathbb R}^2\times{\mathbb I}^2$ (or ${\mathbb T}^2\times{\mathbb I}^2$) described by
$$\left\{
\begin{array}{l}
\dot{\theta}_1 = \omega_1 - ...

**6**

votes

**1**answer

231 views

### Why aren't operator semigroups studied from a dynamical perspective?

Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics.
When studying ...

**3**

votes

**0**answers

103 views

### Approximating solutions of non-linear differential equations

I have met a system of non-linear equations as follows,
$$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$
...

**27**

votes

**2**answers

2k views

### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...

**3**

votes

**2**answers

205 views

### Dynamic of $SL_2(\mathbb{Z}$) on $\mathbb{C}^2$

I imagine the dynamic of $SL(2,\mathbb{Z}$) on $\mathbb{C}^2$ has been studied. Does one know if it is recurrent or ergodic (with respect to the Lebesgue measure) ? Is there any explicit description ...

**2**

votes

**2**answers

159 views

### Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...

**2**

votes

**0**answers

122 views

### Reference for and Properties of the $alpha$-entropy

Let $T : X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...

**6**

votes

**3**answers

292 views

### Poincare recurrence theorem and convergence on compact metric spaces

I am looking for a proof (or a reference to a proof) of the following theorem:
Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure ...

**22**

votes

**3**answers

687 views

### Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
Starting with any triangle $T$, one forms $T'$ by connecting ...

**2**

votes

**1**answer

281 views

### Some puzzles about the three conditions in a paper of D.Berend

Recently, I am reading a paper titled "multi-invariant sets on tori" by D.Berend.
I am puzzled by the three necessary and sufficient conditions given there.
Could you provide me with some concrete ...

**6**

votes

**2**answers

317 views

### Rotation numbers for amenable group actions on the circle

Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim_{n \to \infty} \tilde{f}^n(0)/n) + ...

**2**

votes

**1**answer

172 views

### Is it possible to approximate an area-preserving diffeomorphism by a sequence of conjugates of periodic rotations?

Is it possible to approximate an area-preserving diffeomorphism $T$ of the disk $\mathbb{D}^2$ by a sequence of conjugates of periodic rotations $B_n^{-1} S_{\frac{p_n}{q_n}} B_n$, where $ ...

**0**

votes

**1**answer

118 views

### Stability analysis of ODE

My questions concerns the stability analysis of the following dynamical system :
$\dfrac{d}{dt} a_{i}(t) = D_{i} + \displaystyle{\sum_{j=1}^{n}L_{ij}a_{j}(t) + \sum_{j=1}^{n}\sum_{k=1}^{n} C_{ijk} ...

**4**

votes

**1**answer

172 views

### Under what conditions can interval exchanges be approximated by periodic maps?

Under what conditions can an interval exchange be approximated by periodic maps? (in the weak topology for the Lebesgue measure on $[0,1]$ ).
Are there non-trivial examples of periodically ...

**9**

votes

**0**answers

361 views

### Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...

**5**

votes

**1**answer

206 views

### Dynamics of $3^x$ mod 1

Consider the map $f(x)=3^x$ mod 1. Using the the iterated function system $T_{0}x=\log_{3}(x+1), T_{1}x=\log_{3}(x+2)$ we see that $f$ is dynamical conjugated to a full shift on two symbols. Moreover ...

**3**

votes

**1**answer

156 views

### Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...

**6**

votes

**2**answers

356 views

### “is topologically mixing” vs. “is topologically transitive” in the defition of chaos

This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours.
Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits"
as the ...