Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations.
2
votes
1answer
60 views
Classification of ergodic measures for circle expanding maps
Let us consider the classical self-covering of the circle $S^1=\mathbb{R}/\mathbb{Z}$ given by
$$\times_d(x) = dx \mod 1$$
where the degree $d$ is any integer greater than $1$.
There are a wealth of ...
0
votes
0answers
53 views
Characterization of the stable manifold [on hold]
Assume we study a (finite dimensional) differential system
$$
x'(t)=f(x(t)), \quad x(t) \in \mathbb R^n,
$$
for a smooth function $f$ and such that $0$ is an equilibrium point. Thus, we have existence ...
1
vote
0answers
32 views
Techniques for finding the stationary state of a continuous-state, discrete-time Markov process
I'm interested in a continuous-state, discrete-time Markov process. Let the distribution at time $t$ be $f_t(x)$. The update equation has the form
\begin{equation}
f_{t+1}(x) = \int f_t(x') g(x', x) ...
7
votes
0answers
97 views
Ricocheting pinball-like shot: Complexity?
Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...
2
votes
1answer
109 views
Lyapunov exponent for circle diffeomorphisms
Let $f:S^1\to S^1$ be an orientation-preserving circle diffeomorphism with irrational rotation number (see here). Then the system $(S^1,f)$ admits a unique invariant measure, say $\mu_f$.
Let ...
16
votes
4answers
2k views
A question on Collatz's conjecture
Let $C$ : ${\mathbb N}\longrightarrow {\mathbb N}$ be Collatz's map defined by $C(n) = 3n+1$ if $n$ is odd, and $C(n)=n/2$ if $n$ is even. Then according to Collatz's conjecture, we should have $C^k ...
1
vote
0answers
55 views
Uniqueness of analytic center manifold
In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
1
vote
1answer
74 views
Palis' conjecture and Newhouse's results
Newhouse proved that in the space of C^r smooth diffeomorphisms r > 2, a topologically general dynamical system can have an infinite number of attractors (he goes even further, actually in showing the ...
5
votes
1answer
100 views
Translation surfaces & integer multiples of $\pi$
Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011),
defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...
9
votes
3answers
293 views
Which polygons have *simple* periodic billiard paths?
I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Masur proved in the 1980's that every rational polygon
(vertex angles rational multiples ...
3
votes
0answers
84 views
On the decay of correlations of an ergodic sequence over the set $X_{0}=0$
The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...
3
votes
1answer
131 views
invariant measure of uniquely ergodic horocycle flow
Let $S$ be a compact connected orientable surface of variable negative curvature, and let $M=T^1S$ be the unit tangent bundle of $S$. Then, we know from the paper of Brian Marcus (*) that the negative ...
2
votes
0answers
85 views
Linear forms with best approximation vectors lying in a subspace
Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in ...
1
vote
0answers
50 views
A cohomology associated with a codimension one foliation(2)
What is an example of a codimension one foliation of a manifold for which this cohomology is finite dimension for all dimension $*$?
Moreover what is the description of this cohomology for ...
1
vote
0answers
183 views
A Lie algebra associated with a one dimensional foliation
A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...
1
vote
1answer
81 views
A cohomology associated with a codimension one foliation
Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex:
$$\phi:\Omega^{i}(M)\to ...
1
vote
1answer
50 views
positively invariant set respec to fractional system
In my research I need to show that the set
$$M := \{X \in \mathbb{R}^4,X≥0\}$$
where
$$X(t)=(x_1(t),x_2(t),x_3(t),x_4(t))^T$$
is positively invariant with respect the following system of ...
2
votes
0answers
65 views
Analogous to a PDE but where independent variable is a function
Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...
10
votes
1answer
192 views
Choosing a metric in which homeomorphism is Holder continuous
Let $X$ be a compact metrizable space, and let $f:X \to X$ be a homeomorphism. Is it always possible to choose a compatible metric on $X$ in which $f$ is Holder continuous? I've tried some simple ...
10
votes
2answers
430 views
On the boundary of the twindragon
Let $\mathcal T$ be the famous twindragon, i.e.,
$$
\mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}.
$$
Then, as is well known, $\mathcal T$ has a ...
0
votes
1answer
64 views
Help with notation for the state of a dynamical system defined by a PDE
Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of ...
1
vote
0answers
35 views
Recent Survey on Dynamics of Linear Operator
I'm studying Linear Dynamics using the textbook Linear Chaos by grosse erdmann. I'm looking for a recent encyclopaedic article/survey which gives me a big picture of the area.
It seems erdmann and ...
3
votes
0answers
74 views
Stationarity of Brownian motion with drift
Suppose the following SDE for $X_t$ is well-posed:
$$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$
For what $\Phi\in C^1(R^d)$ will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ ...
3
votes
1answer
72 views
conjugacy between geodesic flows on 2-tori
Let $(T_1,g_1)$ and $(T_2,g_2)$ be two flat tori of dimension 2 such that their geodesic flows are $C^0$-conjugated, is there an isometry between $(T_1,g_1)$ and $(T_2,g_2)$ ?
I emphasize the fact ...
0
votes
0answers
143 views
What is the state of the art of visualizing bifurcations for “difficult” dynamical systems?
This question is related to my other recent question on MO (although I am not confident that the dynamical system described in that other question is actually "difficult," in the sense that I will ...
4
votes
2answers
190 views
The closure of all periodic homeomorphisms of circle
Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...
7
votes
1answer
169 views
Analytic diffeomorphisms of the circle from complex domains
Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms:
$$\phi : (D^2,S^1) \to ...
4
votes
0answers
273 views
A vector space associated with a vector field on a symplectic manifold
Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...
4
votes
0answers
97 views
Unit eigenvalue of the linearized Poincare return map
Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...
5
votes
0answers
150 views
Is Akcoglu's theorem for power bounded positive operators still an open problem?
I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.
" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...
3
votes
1answer
135 views
Is there literature available on iterated function systems of the form $f^n = (g f^{n - 1}, g f^{n - 2}, \ldots)$?
This question is motivated by another question on math.stackexchange.
From a function $g:X^k\to X$ it is possible to define an iterated function system on $X^k$ with the function $f:X^k\to X^k$ ...
0
votes
1answer
173 views
Some quantities which definitions are (somehow) similar to the classical Divergence
Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...
5
votes
0answers
129 views
Derivative of Wronskian
In the proof of Theorem 2 in this paper here on arxiv on page 10 for $k=2$ it is claimed that if the Wronskian of two solutions $y_1,y_2$ to the differential equation
$$-y''_i(x) + q(x) y_i(x) = ...
0
votes
0answers
79 views
Asymptotic pseudo orbit of an action
Let $G$ be finitely generated group (i.e $G= <S>$ $S=\{ s_1, ...,
s_n\}$) and $\varphi:G\times M\longrightarrow M$ is an action then
$f:G\longrightarrow M$ is called $\delta$- pseudo orbit if ...
4
votes
2answers
238 views
Is there any elementary proof of No wandering domain for polynomials
It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps.
I think it is interesting to ask whether we ...
13
votes
1answer
252 views
Nonperiodic points of homeomorphisms of a ball
Suppose $B$ is a $d$-dimensional ball (for some $d \geq 1$) and $T$ is a homeomorphism from $B$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that ...
3
votes
1answer
161 views
invariant measures of the expanding maps on the circle
I would be very happy to know about original references for the following results;
For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1)
(1) There exist ...
1
vote
1answer
175 views
Vector fields whose divergence are proper maps
Let $X$ be a polynomial vector field of degree $2$ on $\mathbb{R}^{2}$. Does there exist a nonvanishing smooth function $g$ such that $Div(gX)$ is a proper map?Or at least the zero locus of ...
1
vote
1answer
63 views
Groups arising as direct limits of a stationary system of primitive matrices over the integers
I am interested in the kinds of groups of the form $\displaystyle\lim_{\longrightarrow}(\mathbf{Z}^k,M)$ where $M$ is a primitive (some power of $M$ has strictly positive components) $k\times k$ ...
5
votes
1answer
147 views
Non-closed geodesics on a convex polyhedron in $\mathbb{R}^3$
Let $P$ be the surface of a closed convex polyhedron in $\mathbb{R}^3$.
Q. Does every non-closed geodesic $\gamma$ fill $P$ densely?
Of course $\gamma$ cannot pass through a vertex of $P$, but ...
10
votes
1answer
299 views
Nonperiodic points of piecewise-linear homeomorphisms
Suppose $K$ is a compact polytope and $T$ is a piecewise-linear homeomorphism from $K$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that ...
1
vote
1answer
169 views
Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle
Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much.
Let $M$ be a n dimensional manifold. ...
3
votes
0answers
60 views
Lorenz attractor power spectrum
If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
0
votes
0answers
54 views
Rational dynamical system with nonnegative paramaters
let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
9
votes
1answer
192 views
Perron-Frobenius theory for reducible matrices
Can someone suggest some sources/references dealing with the Perron-Frobenius theory for nonnegative matrices that are reducible?
Specifically, if $A\ge 0$ is a $d\times d$ matrix with no assumptions ...
4
votes
1answer
166 views
Resolvent of a triangular matrix
Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$?
Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
2
votes
2answers
178 views
Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision ...
0
votes
0answers
43 views
Lifting $SHFC$ to non singular foliations
In this question we would like to complexify the idea in the following post.
We would like to lift a "singular holomorphic foliation by curves", briefly "SHFC", of $\mathbb{C}P^{2}$ to a ...
3
votes
2answers
134 views
The upper and lower bound of the projection of a subshift of finite type
I am thinking a problem: given a subshift of finte type of $\{0,1\}^{\mathbb{N}}$ and $2>q>1$, where $q$ is a real number. Then how can we find the largest and smallest numbers of the projection ...
7
votes
2answers
275 views
Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$
I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
