Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hmiltonian dynamics, ergodic theory.

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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
2answers
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
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3answers
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
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6answers
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
2answers
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
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1answer
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
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0answers
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
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1answer
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
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1answer
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
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2answers
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
1answer
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
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1answer
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
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2answers
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): ...
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0answers
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
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2answers
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
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1answer
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
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0answers
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
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0answers
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
1answer
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
1answer
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
2answers
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
2answers
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
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0answers
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
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0answers
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
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1answer
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
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0answers
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
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1answer
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
1answer
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
2answers
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 ...
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2answers
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
1answer
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 ...
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0answers
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
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0answers
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
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1answer
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
0answers
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
1answer
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
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0answers
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
1answer
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
3answers
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
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0answers
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
1answer
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
0answers
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
3answers
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 ...
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votes
1answer
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
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2answers
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
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1answer
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
2answers
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 ...
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0answers
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
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7answers
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
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1answer
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 $x$ ...