Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
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Special Cases of Duistermaat-Heckman Formula
The Duistermaat Heckman localization formula states how integrals over symplectic spaces with Hamiltonian $U(1)$ group actions.
$$ \int_M \frac{\omega^n}{n!} e^{-\mu} = \sum_{x_i \text{ fixed}} \frac{...
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A question about Mirzakhani et. al.'s algebraicity theorem
While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...
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Fourier transform of x2 invariant measure
Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
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Existence of a vector field with a finite number of limit cycles.
The following question is related to the Seifert conjecture.
Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
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Distribution of $\{cn^a\}$
Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) ...
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An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)
Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question.
Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ(...
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Fixed objects of the M endofunctor on category Meas
Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces.
As Gerald Edgar &...
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Curvatures of stable and unstable manifolds
Let $(M,g)$ be a closed Riemannian manifold and $f:M\to M$ be a $C^r$ ($r\ge2$) Anosov diffeomorphism, that is, there is a continuous hyperbolic splitting $TM=E^s\oplus E^u$ with respect to the ...
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Equivalent definitions of topological weak mixing
A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The ...
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A problem involving the inverse Collatz map
Let $C$ be the Collatz map on the natural numbers, defined by:
$$C(n) :=
\begin{cases}
n/2 & \text{if} \;n \;\text{even} \\
(3n+1)/2 & \text{if} \;n \;\text{odd}
\end{cases}$$
The inverse ...
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Nonconventional ergodic averages for commuting transformations
Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...
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When does a Lagrangian dynamical system have an equivalent Hamiltonian description?
Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$
(i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,......
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Is there a universal $\omega$-limit set?
For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$.
For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...
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How does the mixing time of a geodesic flow on a surface vary with the genus?
I have been looking at the numerical behavior of a particular quantity (of no direct importance here, though if you must know the gory details start with figure 17 here) associated to the geodesic ...
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Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
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Shape rotate, intersect; repeat: disk or empty set?
This question concerns a process that iterates intersection of
randomly rotated planar shapes.
Start with a simply connected region $R_0$ in the plane,
and let $c_0$ be the centroid of $R_0$.
Rotate $...
- 151k
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Centralizers in diffeomorphism groups
Let $M$ be a compact manifold and let $\mathrm{Diff}^{1}(M)$ denote the group of $C^{1}$-diffeomorphisms of $M$. Let $Z(f) := \{g \in \mathrm{Diff}^{1}(M) \mid gf = fg \}$ denote the centralizer of $f$...
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On the first sequence without collinear triple
Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
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Fixed-points of a topological circle action
Suppose the circle group $G = S^1$ acts on $X$.
If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. ...
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when is the Brun continued fraction periodic?
I was hoping to figure this one out on my own. There's this nice paper by Avila on various "subtractive" Euclidean algorithms. Here is one he attributes to Viggo Brun:
$$ (x,y,z) \mapsto \text{sort}...
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Topological entropy and periodic sequences of a subshift
Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$.
...
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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 ...
- 151k
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Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am ...
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Raphael Douady's thesis: Applications du théorème des tores invariants
Raphael Douady's thesis, Applications du théorème des tores invariants, has been cited in numerous papers by many experts.
According Wikipedia, he proves of the equivalence of KAM ...
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Steepest descent/gradient descent as dynamical system
In my research on optimization research, I thought about attempts to bridge gradient descent (deterministic standard) via a dynamical system of sorts, meaning if I look at the iterations of Gradient ...
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amenable equivalence relation generated by an action of a non-amenable group
Question. Give a (possibly elementary) example of a probability measure preserving action $\rho\colon G \curvearrowright X$ of a finitely-generated discrete group $G$ on a standard borel space $X$ ...
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Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$
The group $\mathrm{PSL}_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace \infty\rbrace$....
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Classes of dynamical systems
A consequence of Birkhoff ergodic theorem tells us that ergodicity is equivalent to:
$\forall A,B \in \mathcal{B} \quad \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to \infty}{\...
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Existence of nonergodic polygonal billiard
Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one.
A standard conjecture is that a ...
- 2,696
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How to understand a solenoid?
Consider the invertible extension of the circle-doubling map $T(x)=2x \pmod 1$, the new system can be represented as $X=\{(x_k)\ | \ x_{k+1}=T(x_k)\}$ (see GTM 259 M. Eindiedler & T. Ward ...
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What does the flow of the principal symbol of the differential operator tell us about the PDE?
Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
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Compressible Ebin-Marsden?
In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of ...
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An algorithm for Poincare recurrence time
Define the function $[0,+\infty) \rightarrow R$:
$$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$
I want a number $t $ bigger than $10^7$ such that
$$ f(t) > 4 -...
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Knots and Dynamics. Recent breakthroughs?
I recently started reading Étienne Ghys slides on knots and dynamics <http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf> which seem very interesting. I know this approach to knots and dynamics is ...
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5n+1 sequence starting at 7
Consider the following variant of the Collatz function: $f:\mathbb N\rightarrow\mathbb N$ is defined by
\begin{equation}
f(n):=\begin{cases}
n/2 & \text{if $n$ is even}\\
5n+1 & \...
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Does a compact nonflat surface without conjugate points have ergodic geodesic flow?
I read this as a conjecture in the paper by Ballmann-Brin-Burns, titled "On Surfaces with No Conjugate points" JDG 25(249-273), 1987.
What is current status of this conjecture?
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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, ...
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In a non-compact metric space, topological transitivity need not imply onto
I had asked this question on Mathematics Stack Exchange yesterday but it got no response so I'm asking here.
Let $X$ be a compact metric space and $f:X \to X$ be continuous. If $f$ is topologically ...
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is the geodesic flow on Hyperbolic Plane completely integrable?
I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and $H:M\...
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Bertrand theorem - central forces
Here is a version of Bertrand theorem. Let us consider a force $F(r)$ which depends only on the distance to a given point. If all trajectories which remain bounded are closed, then either $F(r)=ar$ ...
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What do singular, atomless invariant measures of $\times d$ look like?
Consider the circle map $\times d:x\mapsto dx \mod 1$. The lebesgue measure is the only absolutely continuous invariant probability measure, but this map has many other invariant measures. Of course, ...
- 14.4k
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Dimension of orbit versus invariant functions
$\def\CC{\mathbb{C}}$Let $K = \CC(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$; in the cases I care about, $G \cong \mathbb{Z}$. Then the field of $G$-invariants, $K^G$, ...
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Intuition of Kolmogorov-Sinai entropy
For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself.
For each finite measurable partition $\mathcal{A}=\{A_i\}_{i=1}^{m}$ of $X$, we can define
$h(\mathcal{A},...
- 1,706
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Homomorphisms from $\mathbb{R}$ to $\mathrm{Homeo}^+(\mathbb{R})$, or "fractional iterations"
Let $G$ be the group of orientation-preserving homeomorphisms (or, if you prefer, diffeomorphisms) of the real line. Does there exist a natural way to associate, to each function $f \in G$, a ...
- 7,328
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Topological objects associated to Steinerberger's 4-regular graphs
Very recently, in arXiv:2008.01153, Steinerberger has associated to any sequence $(x_n)_{n\in\mathbb{N}}$ of distinct real numbers a 4-regular graph.
In the case irrational multiples, like $x_n=n\sqrt{...
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What is the current status on methods to find limit cycles?
What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form:
$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$
Where $...
- 117
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Interior periodic points of area preserving homeomorphisms of a pair of pants
A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...
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Persistence of fixed points under perturbation in dynamical systems
Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs). Assuming that the system has a finite set of fixed points, I am interested in knowing (or obtaining references about) ...
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A regularity property of transition matrices for the cat map
I've noticed a rather strange phenomenon (not important for my particular research, but interesting) and wouldn't be surprised if someone more versed in symbolic dynamics (i.e., just about anyone who ...
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Suggested reading for thermodynamic formalism
Are there any good books out there that can serve as an introduction to thermodynamical formalism in dynamical systems?
I know only Zinsmeister's short "Thermodynamical formalism and holomorphic ...
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