**1**

vote

**1**answer

86 views

### On solution of a recursion with rectangular matrices

Greetings to members here.
The question is how to calculate the solution $S(k)$ of the following recursive equation
$$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$
where $J$ and $A$ are rectangular not ...

**27**

votes

**4**answers

724 views

### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be ...

**3**

votes

**0**answers

59 views

### Do identical orbit tiles imply identical combinatorial types?

Given a periodic trajectory on a triangle, we can associate to this trajectory a sequence of integers $1,2$ and $3$ by labeling the edges of the triangle and taking the sequence of edges the ...

**0**

votes

**1**answer

203 views

### Recurrence and transience of cocycle over a dynamical system

Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and ...

**19**

votes

**4**answers

941 views

### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...

**8**

votes

**1**answer

568 views

### What is known about the strong Arnold conjecture?

Here are the two versions of Arnold's conjecture on Hamiltonian orbits:
Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate ...

**1**

vote

**2**answers

272 views

### complex dynamics in several variables

Dear mathematicians,
I want to know how much advance there has been in complex dynamics of several variables. I am at present reading Carleson's book on Complex Dynamics on one variables.Curious to ...

**4**

votes

**2**answers

164 views

### Lyapunov exponents for PDEs

How can one define and calculate (analytically or numerically) Lyapunov exponents for partial differential equations? Do there exist examples of nonlinear PDEs for which Lyapunov exponents can be ...

**3**

votes

**1**answer

118 views

### The relations between conservative part and conservativity

I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for ...

**6**

votes

**2**answers

267 views

### Iterates converging to a continuous map

I have no doubt that the following observation is quite well known. Let $\varphi:[0,1]\to [0,1]$ be a continuous map. Assume that the iterates $\varphi^n$ converge pointwise to some continuous map ...

**1**

vote

**0**answers

108 views

### Adding a damping term to a dynamical system or Markov process: what happens to invariant measures?

Consider the continuous-time Markov process on ${\mathbb R}^n$ described by the SDE
$\dot{x}(t) = F(x(t)) + \xi(t)$
where $F:{\mathbb R}^n \to {\mathbb R}^n$ is a smooth mapping, and $\xi(t)$ is a ...

**4**

votes

**2**answers

129 views

### Chain Recurrent Set of a Isometry

Let be $T:X\to X$ a topological dynamical system, $X$ a compact space and $T$ is also a isometry. Let be $\mathcal{R}(T)$ the chain recurrent set of $T$.
Theorem: $\mathcal{R}(T)=X$
There is a ...

**1**

vote

**2**answers

295 views

### Replacing large-dimensional ODE systems with one PDE [closed]

Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?

**2**

votes

**0**answers

165 views

### Is it true that a solid, minihedral cone in infinite dimensions cannot be regular?

Background
Consider a real Banach$^1$ space $V$. We'll call a subset $V_+ \subseteq V$ a cone if
$V$ is closed,
$\alpha V_+ \subseteq V_+$ for every $\alpha \geqslant 0$ and
$V_+ \cap (-V_+) = ...

**8**

votes

**0**answers

274 views

### Reference - Asymptotic geodesics on compact surfaces without conjugate points

I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of ...

**15**

votes

**10**answers

2k views

### Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.)
I am an undergrad in math ...

**1**

vote

**1**answer

341 views

### Partial linearization near a hyperbolic fixed point--Classical scattering

I am currently reading the famous article "Universal Properties of Maps on an Interval"
by Collet, Eckmann and Lanford related to the Feigenbaum-Coullet-Tresser universality.
I am in particular ...

**5**

votes

**2**answers

208 views

### Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...

**2**

votes

**2**answers

226 views

### Equivalence of two definitions of Lyapunov exponents

I saw in articles two different definitions for Lyapunov exponents of a discrete dynamical system.
Let's consider a discrete dynamical system
$$
x_{k+1}=f(x_{k}),\quad x_{k}\in\mathbb{R}^{n},\quad ...

**3**

votes

**0**answers

83 views

### Calculation of Lyapunov exponents for infinite systems of differential equations

Can you give an example of a function $\varphi$ and sequences $\{b_{i}\}$ and $\{a_{ij}\}$
for which one can calculate Lyapunov exponents of such the infinite system of differential equations
and ...

**17**

votes

**1**answer

493 views

### Can the expansion of a large integer in all bases consist of almost all zeroes?

Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of non-zero digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the ...

**1**

vote

**1**answer

347 views

### Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...

**4**

votes

**2**answers

334 views

### Algebraicity of the “outer” boundary of the Mandelbrot set

Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as
$$
t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu ...

**1**

vote

**3**answers

183 views

### Applied examples of (non)uniformly hyperbolic and/or ergodic systems

I try to give reference to completely applied examples of (non)uniformly hyperbolic and/or ergodic systems. With completely applied I don't mean an irrational rotation on the torus but from other ...

**4**

votes

**1**answer

260 views

### Characterising ergodicity of continuous maps

Hello all.
Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is not ergodic.
Does there necessarily exist a Borel ...

**1**

vote

**0**answers

119 views

### Detecting Non-Transversality

Suppose $f \colon \mathbb{R}^n \to \mathbb{R}$ is Morse and has finitely many critical points. Is there an algorithm for determining whether there exists a saddle-saddle connection (an orbit of grad ...

**1**

vote

**1**answer

238 views

### Derivative of a random process

Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t.
I consider applying a (stochastic)derivative operation to the random process. What is the ...

**2**

votes

**6**answers

313 views

### Applications of discrete-time dynamics

Hello,
I am a graduate student in the field of discrete-time dynamics. I am wondering about applications of this field outside of mathematics. More precisely, I would like to know if there are "real ...

**5**

votes

**2**answers

230 views

### pointwise ergodic theorem and mean sojourn time

Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic ...

**3**

votes

**0**answers

130 views

### Perturbation of Morse functions at critical points leaving stable manifolds invariant

Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858).
Is there a way ...

**6**

votes

**0**answers

216 views

### Do ergodic isometries have discrete spectrum?

Let $X$ be a metric space, $\mu$ a Borel probability measure, and
$T:X\rightarrow X$ be an ergodic measure preserving isometry.
Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...

**1**

vote

**1**answer

93 views

### Reference Request: Structural Stability of Gradient Fields

I am asking for a reference that contains a proof of Theorem 4, which is on page 315 of the following text:
Hirsch, Morris W., and Stephen Smale.
Differential equations, dynamical systems, and ...

**1**

vote

**1**answer

209 views

### Integration by parts wrt. a Morse function on its basin of attraction

Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$
$$
\forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad ...

**8**

votes

**2**answers

349 views

### Birational Automorphisms and infinite divisibility

Suppose $X$ is some algebraic variety. It can be over $\mathbb{C}$, but it doesn't have to (but char $0$ preferred).
Is it possible that the additive group $\mathbb{Q}$ acts on it birationally, ...

**2**

votes

**0**answers

180 views

### Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...

**1**

vote

**0**answers

150 views

### Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets.
He proved in Theorem 8.8 that in each finite partition of ...

**10**

votes

**2**answers

542 views

### The height of an orbit under rational self-maps

I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I ...

**1**

vote

**0**answers

187 views

### Interesting examples of minimal action on torus

Edit 1:This is a cross post on MSE. See math.stackexchange.com/q/289595/12952
Edit 2:I originally asked for finite group actions as I thought that will be easier. But as pointed out by Victor minimal ...

**0**

votes

**0**answers

97 views

### Degree of freedom restricted by inequalities

Motivational example
Consider a polyhedral graph $G$. A realization of $G$ is given by a convex polyhedron which is - essentially - characterized by the angles between the edges emanating from each ...

**5**

votes

**0**answers

161 views

### Fibre Mixing for Dynamical Systems

Hi all,
I'm interested in understanding a fairly difficult theorem of Lindenstrauss Peres and Schlag. In that paper the authors prove that certain dynamical systems related to beta expansions and ...

**5**

votes

**1**answer

467 views

### “Explicit” examples of Irrational numbers very well approximated by rationnal numbers

This question relates to this one and that one.
Some background
In the setting of discrete holomorphic dynamics (say, Julia sets)
an irrational $\lambda$ is said to be well approximated by rational
...

**3**

votes

**1**answer

228 views

### A theory of bifurcation of braids ?

I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional ...

**1**

vote

**2**answers

123 views

### Relation between entropy of one-parameter group and single elements of this group

My question is motivated by the hypothesis of the Lindenstrauss' proof of arithmetic quantum unique ergodicity, and the answer to my question is certainly known. However, I could not find it in the ...

**7**

votes

**2**answers

440 views

### Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory:
Invariant Measures
Poincaré recurrence Theorem
Invariant Measure For Continuous Transformations
The Ergodic Theorems and Applications
Ergodic ...

**0**

votes

**3**answers

152 views

### Conley index for isolated invariant sets with no exit points

Conlay described in $\textit{Isolated Invariant Sets and the Morse Index (1976)}$ the bases of what would be known as Conley Index Theory.
For the sake of simplicity let's think of vector fields ...

**10**

votes

**1**answer

289 views

### Does the $n$-gonal billiards conjecture follow from the $m$-gonal conjecture when $m>n$?

For $n\ge 3$, define the $n$-gonal billiards conjecture as the statement
All convex $n$-gons admit periodic billiard trajectories.
To the best of my knowledge this question remains open for all ...

**1**

vote

**1**answer

258 views

### Generalized Eigenvector in Dynamical System in Infinite Dimensions

Consider a system of linear delay differential equations:
$$
\dot{z_1}(t) = z_1(t) + z_2(t-1)
$$
$$
\dot{z_2}(t) = z_2(t) + z_3(t-1)
$$
$$
\dot{z_3}(t) = z_3(t) - z_1(t-1)
$$
The ...

**2**

votes

**1**answer

64 views

### (A)periodicity and (In)dependence on the boundary condition for optimization problem related to ODE

The question is pair to MO117505 and translates some problem on error-correction codes to similar problem about differential operators. (See also If “force” is periodic does it imply “velocity” is ...

**1**

vote

**1**answer

108 views

### (A)periodicity and (In)dependence on the boundary condition for some discrete analog of ODE (convolutional codes)

(See also MO117508, MO116611). This post describes somewhat real problem with convolutional codes. Let me first try to give brief and vague formulation of the question, later give details.
Problem ...

**1**

vote

**1**answer

191 views

### Terminology question in dynamical systems

Let $X$ be a topological space and let $f:X\rightarrow X$ be a continuous self-morphism of topological spaces. Let $Y$ be a closed $f$-stable subset of $X$, that is, suppose $f(Y)\subseteq Y$. ...