**0**

votes

**0**answers

32 views

### Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space.
I am interested in informative examples and applications of such systems. I know ...

**2**

votes

**2**answers

144 views

### Question on the number of equilibria

Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or ...

**27**

votes

**3**answers

973 views

### Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...

**18**

votes

**6**answers

2k views

### Parametrization of the boundary of the Mandelbrot set

Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.
The ...

**10**

votes

**2**answers

414 views

### Blinking graphs

For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a bit assignment for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the ...

**2**

votes

**1**answer

95 views

### Discrete spectrum and almost periodicity

According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to ...

**5**

votes

**0**answers

67 views

### General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics.
I have been interested in some technical properties of the ...

**3**

votes

**1**answer

137 views

### Transformation extending all ergodic rotations

Is there an invertible measure-preserving transformation (preferably a nice one) admitting every irrational rotation as a factor ? I guess the spectrum is the relevant tool to address this question ...

**22**

votes

**1**answer

1k 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 ...

**14**

votes

**2**answers

593 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 ...

**6**

votes

**1**answer

736 views

### Simple example of renormalization

As far as I understand, the RG theory, or functional RG theory is a mathematical tool for moving in the "scale dimension". The tool can be used for calculation of Feigenbaums constant (e.g. mentioned ...

**4**

votes

**1**answer

175 views

### What are the generating partitions of the odometer?

According to the countable generator theorem, every ergodic invertible measure-preserving transformation has a generating partition.
What are the generating partitions of the dyadic odometer ? I ...

**9**

votes

**2**answers

910 views

### Integrable dynamical system - relation to elliptic curves

From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (http://en.wikipedia.org/wiki/Integrable_system):
...

**0**

votes

**0**answers

55 views

### Is the Rossler attractor globally stable?

The standard form of the Rossler system is $\frac{dx}{dt}=-y-z$, $\frac{dy}{dt}=-x+ay$, $\frac{dz}{dt}=b+z(x-c)$. For simplicity, consider the well known chaotic attractor that exists at a=b=0.2, ...

**1**

vote

**2**answers

73 views

### Transformations whose product with a given rotation are ergodic

I am interested in the ergodic (invertible) transformations $T$ such that $T\times R_\theta$ is ergodic where $R_\theta$ is the rotation on $S^1$ with a given irrational angle $\theta$ (not all ...

**13**

votes

**1**answer

515 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**0**

votes

**0**answers

33 views

### Transformations whose product with the odometer are ergodic

Let $T$ be an invertible ergodic transformation on a Lebesgue space $X$ and $O$ be the dyadic odometer on $(0,1)$. Is it true that $T\times O$ is ergodic if and only if $T^{2^n}$ is ergodic for every ...

**2**

votes

**0**answers

44 views

### Techniques for the analysis of interacting particle systems with a finite number of particles, which do not resort to limiting arguments?

I am interested in pointers to (keywords/authors) recent research on the analysis of interacting particle systems with a finite number of particles which do not resort to limiting arguments converting ...

**6**

votes

**1**answer

82 views

### Sign problem in a Calogero-Moser system: proof of integrability?

Everyone of us had sometimes this awful feeling that some sign is lost in a calculation and that this sign is perturbing some fundamental understanding of what is going on. I feel the same has ...

**12**

votes

**1**answer

481 views

### Example of a quasi-Bernoulli measure which is not Gibbs?

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only.
A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$,
$$
C^{-1} ...

**10**

votes

**2**answers

403 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 ...

**3**

votes

**2**answers

1k views

### Invariance of dynamical system under a transformation

I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out why this is happening (for quite some time now actually). Any help ...

**0**

votes

**0**answers

33 views

### Sufficient condition for a differential inclusion to have a global attractor

Let $\dot{x}(t) \in f(x(t))$ be a differential inclusion (d.i.). Is there any sufficient condition under which it will have a singleton global attractor ?
We know the condition if $f$ is single ...

**4**

votes

**0**answers

135 views

### Dynamical Mordell-Lang on Kahler manifolds?

Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...

**8**

votes

**1**answer

314 views

### resampling over Bowen balls

Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...

**3**

votes

**0**answers

74 views

### Reconstructing a vector field on the circle

Consider a ODE on the circle of the form
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \omega(x) \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix} \begin{pmatrix} x_1 \\ ...

**3**

votes

**1**answer

106 views

### Packing measure and Kleinian groups

There has been "some" debate on the notion of fractal (as an illustration, see for example the discussion in this link). One of the possible notions includes relating Hausdorff dimension and packing ...

**11**

votes

**1**answer

219 views

### Unusual digit sets that allow finite expansions for all (positive and negative) integers

Informal introduction
(If you don't like informal introductions, please skip to 'Mathematical formulation')
Whenever our 'decimal positional system' for writing numbers comes up in conversation, ...

**0**

votes

**2**answers

164 views

### If Non wandering Set is whole space then recurrent set is dense?

I tried to prove following statement and use some techniques but I couldn't get result :
Question: If Non wandering Set is whole space then Recurent set is dense??
when $T:X \to X$ is ...

**1**

vote

**2**answers

106 views

### What are good references for spatial dynamics?

Hello I have started working on my PhD a short while ago and wondered if there might be any good introductions to spatial dynamics.
I have a basic understanding of dynamical systems but would like to ...

**-1**

votes

**1**answer

173 views

### Given $f(g(x))$ is convergent, what can be said about the convergence of $f(x)$ and $g(x)$? [closed]

Consider the iterated function $f^n(x)=f(f(f(...f(x))...))$ where $n \in \mathbb Z$ and $f(x)$ is convergent. Then the iterates of $f(x)$ such as $f^2(x), f^3(x), f^4(x)$ are convergent. Now let $s ...

**0**

votes

**0**answers

72 views

### The growth rate of almost periods for almost periodic function

A subset $A \subset \mathbb{R}^2$ is relative dense if there exists $L>0$ such that for every $p\in \mathbb{R}^2$ there exists $p' \in A$ such that $|p-p'|<L.$
A continuous function $f : ...

**0**

votes

**0**answers

60 views

### Joint point of coarse geometry and dynamical system?

My major interest is on dynamical systems,
but I did REU in a coarse embedding problem.
I wonder whether there's some significant connection between those two subjects.
I've tried to google for a ...

**16**

votes

**1**answer

651 views

### Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let
$\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals.
Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := ...

**7**

votes

**0**answers

233 views

### Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...

**4**

votes

**1**answer

126 views

### Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation.
Consider a measurable set $A\subset X$ with $0<\mu(A)<1$
For each $N$ define ...

**5**

votes

**0**answers

91 views

### Topologically transitive, pointwise minimal systems

I'm cross-posting this from SE.
Let $T$ be a group, and let $(X,T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically ...

**1**

vote

**1**answer

169 views

### A quantitative Kronecker theorem

I encounter the following question.
$\textbf{Problem}$: For almost all Matrix $M\in\mathcal M_{m\times n}(\mathbb R),$ all $y\in \mathbb R^m$ and any $N$, small $\epsilon>0$, there exists a ...

**9**

votes

**3**answers

584 views

### Integer dynamics hitting infinitely many primes

I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). To make it specific, ...

**1**

vote

**0**answers

90 views

### Strict factor of a dynamical system with the same entropy [closed]

Say that a factor of an invertible measure-preserving transformation $T$ is strict if it is not isomorphic to $T$. Does there exist an invertibe mpt $T$ such that $0 < h(T) < \infty$ and ...

**6**

votes

**1**answer

114 views

### Generator determined by finitely many translates implies zero entropy

Let $T$ be a measure preserving transformation of a standard probability space $(X,\mathcal{B},\mu)$. A partition $\alpha$ of $X$ is said to be a generator for $T$ if the smallest $T$ invariant ...

**6**

votes

**0**answers

80 views

### How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
...

**5**

votes

**3**answers

592 views

### Poincaré Recurrence and Dense Sets

This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The ...

**-3**

votes

**1**answer

70 views

### Exponential map and hyperbolic invariant set [closed]

Let $(M,g)$ be a Riemannian manifold, and let $f\colon M\to M$ be a $C^1$ diffeomorphism.
Let $p$ be a hyperbolic fixed point of $f$ such that $T_pM= E^s_p + E^u_p$ and $\exp_p \colon T_pM \rightarrow ...

**7**

votes

**2**answers

287 views

### Periodicity in iterated powers of sin, cos, exp

Given a complex number $z$, consider the sequence
\begin{align*}
a_0 & = 1\\
a_1 & = (cos(1))^z\\
a_n & = (cos(a_{n-1}))^z
\end{align*}
This question is about trying to understand ...

**0**

votes

**1**answer

54 views

### Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$

This question has been asked here but there is no answer:
http://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
Consider autonomous ODE $y' = ...

**6**

votes

**2**answers

879 views

### Relation between Hausdorff dimension and Bowen's equation

I am reading the paper Hausdorff dimension for Horseshoes, by McCluskey and Manning. In the following theorem
Theorem:
Let $\Lambda$ be a basic set for a $C^1$ axiom A diffeomorphism $f:M^2\to ...

**0**

votes

**0**answers

179 views

### Can we find an upper bound?

Let $f\in C^1(\mathbb R)$ with $f(0)=0$ and $|f'(x)|\le m$, where $m\in (1,2]$.
Let $x(0)\in\mathbb R$ be arbitrary, and define $x(n),y(n)$ recursively by
$$
x(n+1)=f(x(n)) , \quad\quad
...

**23**

votes

**7**answers

2k views

### If you were to axiomatize the notion of entropy …

What are the axioms that a good notion of entropy must satisfy? Please note that I am not asking for the definitions of various types of entropy such as topological entropy or measure-theoretic ...

**4**

votes

**1**answer

204 views

### What is known about first return times to Markov partitions for Anosov diffeomorphisms?

Consider an Anosov diffeomorphism $T: M \rightarrow M$ and a corresponding Markov partition $\mathcal{R}$ of $M$. For $x \in M$, let $\mathcal{R}(x)$ denote the element of $\mathcal{R}$ containing $x$ ...