Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory, topological dynamics.
2,571 questions
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What literature can I read about the Janibekov effect and the intermediate axis theorem?
I have been studying mathematics for 2 years, and I have already read Terence Tao's publication. Please suggest books on related topics, such as Euler's equations, mathematical modeling, mathematical ...
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1
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Do all primes $>2$ hit $5$?
$2$ is a fixed point of the iteration:
$$q_{n+1}:=\min_{p|(q_n-1)^2+1} p$$
Start with $q_1>2$ prime. Does this iteration hit $5$? (min runs over primes)
- 3,761
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votes
1
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Possible asymptotic behavior of recurrence function
I was wondering and tried to find what are the results known related to recurrence function of a minimal subshift $\Omega \subseteq A^{\mathbb{Z}}$, where $A$ is finite non empty subset.
If I am not ...
- 755
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Book recommendation for smooth ergodic theory
I'm interested in smooth ergodic theory.
Please teach me some recommended books for it.
Actually, now I have been reading the supplement of Katok's book, Introduction to the Modern Theory of Dynamical ...
- 81
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0
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Dynamics of the arithmetic–derivative family $f_k(n)=n+k(D(n)-1)$
Let $D(n)$ be the arithmetic derivative, defined by: $D(p)=1$ for primes $p$, $D(ab)=D(a)b+aD(b).$
For a fixed integer $k$, consider the dynamical system
$$f_k(n)=n+k(D(n)−1).$$
I am interested in the ...
- 1,018
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votes
1
answer
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Does the airplane Julia set contain true circles?
The well known "airplane" Julia set looks like it contains a true circle. To be precise, let $c$ be the real root of $x^3+2x^2+x+1=0$. i.e., $c\approx -1.75$. The Julia set of $z^2+c$ is the ...
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Unprovable statements and generic properties
I should start with the following disclaimer that I know virtually no logic, sorry forgive me if my questions are ill-posed. I appreciate that all of this is probably completely obvious to a logician, ...
- 3,108
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Ergodicity in the Wiener-Wintner Ergodic Theorem [cross-post from MSE]
I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
- 61
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Can $D-D$ be a set of $2$-topological recurrence if $D$ is lacunary?
Background.
For $k \in \mathbb{N}=\{1,2,3,\dots\}$, a set $R \subseteq \mathbb{N}$ is a set of $k$-topological recurrence if for every minimal topological dynamical system $(X,T)$ and every nonempty ...
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Why Isotopic Markings Define the Same Point in Teichmüller Space
Let $S_g$ be a compact orientable surface of genus $g \geq 2$. The Teichmüller space $\mathcal{T}(S_g)$ is defined as the set of equivalence classes of pairs $(X, f)$, where:
$X$ is a Riemann surface,...
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How can one define the Lie bracket of two foliations?
In this question I am inspired by a recently closed MO question who tried to define a kind of Lie bracket on the space of 1 dimensional singular foliations.
However that idea had a gap but I think the ...
- 330
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Density of orbits of action on the boundary of a convex set
Let $X$ be a compact convex subset of the plane. Let $c_1$, $c_2$ and $c_3$ be non-collinear points in the interior of $X$.
For every point $x$ on the boundary $\partial X$, you can draw the line from ...
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Extreme points of a certain compact convex set
Say $\mathfrak{A}$ is a seperable $C^*$ algebra, the space $K$ of states on $\mathfrak{A}$ is compact and convex. Let $\Gamma$ be a countable discrete group acting on $\mathfrak{A}$ via $*$-hom. This ...
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Confusion about the definition of homogeneous orbits from Ratner's theorem
I was reading Ratner's Raghunathan’s topological conjecture and distributions of unipotent flows and confused about a definition in the first page: Ratner used the right action $\Gamma \backslash G$ ...
- 495
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Finiteness of cycles for $T_k(n):=\operatorname{rad}\bigl(\sigma^{\circ k}(n)\bigr)$ when $k$ is fixed
$\DeclareMathOperator\rad{rad}$Let $\sigma(n)=\sum_{d\mid n} d$ be the sum-of-divisors function, and let $\rad(m)=\prod_{p\mid m}p$ be the radical (with $\rad(1)=1$). For a fixed integer $k\ge 1$, ...
7
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2
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Topological full groups
I am trying to learn about topological full groups to do a masters dissertation on this, and am trying to find a solid path to do so. I do not have any C* algebra or dynamics background. What would be ...
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107
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The problem of find the precise $\alpha$-Hölder expoent for some kind of PDE solution
Actually, i would like to know how important is find the precise expoent $\alpha \in (0,1]$, for some Hölder space. At the moment, i get only one example from the paper An intrinsic Liouville theorem
...
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190
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"Finiteness theorem for Limit cycles"
It is well known that every polynomial vector field $\begin{cases} \dot{x}=P(x,y)\\\dot{y}=Q(x,y) \end{cases}$ on $\mathbb{R}^2$ has finitely many limit cycles.See this famous result by Y. ...
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Existence of an Integer Sequence with Specific Properties Related to Divisors and Fractional Parts
Motivated by a problem concerning the existence of quasi-periodic solutions in dynamical systems, I encountered a number-theoretic question that appears somewhat unusual. I would like to know whether ...
- 201
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131
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Finding a closed form solution for a dynamics problem
Recently, I've been working on a dynamics problem defined as follows. Given
$$X(1) = 1,\quad Y(1) = 1,$$
for $t > 1$, let $(X(t), Y(t))$ evolve according to
$$Z(t) = \frac{Y(t) + X(t)/2}{Y(t) + 1},\...
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152
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Invariant sets of foliations
Definition: Let $\mathcal{F}$ be a regular holomorphic foliation on a complex manifold $M$. We say that a subset $ A \subseteq M $ is invariant for $\mathcal{F}$ if $A = \bigcup\limits_{x \in A} L_x $...
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Realizing a Laurent polynomial as a dynamical zeta function or characteristic polynomial from an isolated surface singularity
Given a Laurent polynomial $\Delta(t) \in \mathbb{Z}[t, t^{-1}]$ satisfying:
$\Delta(1) = \pm 1$,
$\Delta(t^{-1}) = t^{\pm k} \Delta(t)$ (symmetry up to a unit),
can $\Delta(t)$ be realized as a ...
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375
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A foliation version of the Seifert conjecture
Edit:
Acording to comment of ThiKu I revise the question as follows:
Seifert conjecture asserts that every smooth or analytic vector field on $S^3$ possesses a closed orbit.
In this question I am ...
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102
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Primitive $S$-adic representations of minimal and linearly recurrent subshifts
I've been looking at the paper Beyond substitutive dynamical systems: S-adic expansions, and was wondering about Theorem 5.2 and Theorem 5.4, dealing with uniformly recurrent and linearly recurrent ...
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Do $x\mapsto e^x-1$ and $x\mapsto x^2$ generate a free group on $\mathbb{R}^+\to \mathbb{R}^+$?
Let $f: \mathbb{R}^+\to \mathbb{R}^+, \ x\mapsto e^x-1$ and $g: \mathbb{R}^+\to \mathbb{R}^+, \ x\mapsto x^2$.
Is the group generated by $f$ and $g$ under composition free? (That is, no non-trivial ...
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Foliation of $TM$ arising from a flat connection
Let $M$ be a $n$ dimensional manifold equipped with a flat connection $\nabla$. Hence the corresponding Ehresmann distribution on $TM$ is integrable so $TM$ is foliated by $n$-dimensional leaves.
To ...
- 330
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73
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Regularity of pseudo-Anosov flows near singular orbits
Let $f: \Sigma \to \Sigma$ be a pseudo-Anosov homeomorphism of a closed surface $\Sigma$. Let $V$ be the vector field on the mapping torus $M_f$ generating the suspension flow. I understand that there ...
- 259
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Minimal composition of time one flows
Consider the Cauchy ODE system
$$\begin{cases}
\dot{x}_t = V(x_t) \\
x_{t=0} = x_0
\end{cases}$$
where $V: \mathbf{R}^d \rightarrow \mathbf{R}^d$ is a compactly supported Lipschitz (and not ...
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Limit cycles with a globally real Jacobian spectrum?
Suppose we have a smooth dynamical system $\dot{x} = f(x)$ on $\mathbb{R}^n$, with the strong condition that its Jacobian $Df(x)$ has only real eigenvalues for all $x \in \mathbb{R}^n$.
Can such a ...
3
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1
answer
108
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Reference Request: accessible points of Wada domain boundaries in $\mathbb R^d$
Say that a compact $K\subset\mathbb R^d$ has the Wada property if there are disjoint, connected open sets $V_1,V_2,\ldots,V_n\subset\mathbb R^d$, $n\geq3$, such that $K=\partial V_1=\partial V_2=\...
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Infinite history in time discrete dynamical systems on complete metric spaces
Consider a continuous map $f:X\to X$ on a non-empty complete metric space $(X,d)$ and its iterates $f^n=f\circ\cdots\circ f$. I am interested in the points $x_0\in X$ with infinite history, i.e., ...
4
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1
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Ghost cycles in dynamical systems
I'm reading a paper under the title Ghost channels and ghost cycles guiding long transients in dynamical systems, here or here.
The authors state that by examining simple ordinary differential ...
user571116
3
votes
1
answer
141
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Reference for a result on chain recurrence
I'd like a reference for this result, which I'm sure is well known among dynamical systems people:
In a compact topological dynamical system $(X,f)$, if $x$ is in chain relation with $y$, then, given ...
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15
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1
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692
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Dynamics from iterated averaging
I've been thinking about structural aspects of conditional expectations. This has resulted in the following surprising approximation result for measurable automorphisms.
Let $(X,\Sigma,\mu)$ be a ...
- 6,876
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2
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412
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Why does the spectral gap tame a $K$-Lipschitz non-linearity on graphs?
I'm working on a problem involving states on a graph, where node states are transformed by a non-linear function. I need to verify a specific inequality related to the graph's spectral properties.
...
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0
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162
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What is the name of graphs that apply functions along their edges?
I am looking for the standard terminology for a mathematical structure consisting of a directed graph where each edge "applies a function passing through it".
Formal Definition
Let $G=(V, E)$...
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0
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How do we prove 1) existence and uniqueness of the global positive solution and 2) extinction for this stochastic system?
Suppose we have the following system:
\begin{equation}
\begin{aligned}
dS &= \Big[\Lambda -\beta_1 S L_2 -\beta_2 S I - \beta_3 S A - \nu S\Big]dt +\sigma_1 SdB_1(t),\\[2ex]
dL_1 &= \Big[p \...
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Is there a generalized version of monotone dynamical system for non smooth dynamics?
I'm aware of the theory of cooperative dynamical systems, but I want to know if there are generalizations. I recall the main notion:
Let $\mathcal{X} \subset \mathbb{R}^N$. A dynamical system $\dot{x}...
2
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2
answers
247
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Proof of convergence of replicator type II dynamics
One of the evolution equations used in evolutionary game theory is the Replicator type II dynamics
$$
x_i(t+1)=x_i(t)\frac{(Ax(t))_i}{x^T(t)Ax(t)}
$$
where $A$ is an $M\times M$ payoff matrix with ...
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161
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Renormalization in complex dynamics
I am a masters student, currently reading about complex dynamics and in particular, the dynamics of polynomials.
Recently, I came across the notion of renormalization. The quadratic-like type, Douady-...
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4
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1
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159
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Any example of a non-equicontinuous system which is the union of all its equicontinuous minimal subsystems
I have asked this question for two days in MSE, see here, but no response so far.
Let $(X,T)$ be a topological dynamical system, consisting of a compact metric space $X$ and a homeomorphism $T: X \to ...
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Peer reviewed works relating spectral graph theory and dynamical systems through piecewise isometries (PWIs)
While Piecewise Isometries (PWIs) represent an area of active research for their insights into complex orbits of dynamical and fractal systems, the study of holonomy-twisted Ihara zeta functions has ...
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77
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Minimal finite-edit shadowing distance in the full two-shift
Let $\Sigma_2 = \{0,1\}^{\mathbb{Z}}$ be the full two-shift with left-shift map $\sigma$ and the standard product metric
$$d(x,y) = 2^{-\inf\{|n| : x_n \neq y_n\}}.$$
Fix $\varepsilon = 2^{-m}$ for ...
0
votes
1
answer
133
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Kneading theory on the tree map
As is well-known, Kneading theory serves as a fundamental and powerful tool in the study of dynamical systems. I am curious whether there exists any connection between Kneading theory and tree maps. ...
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333
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Countable-state shift spaces with greater measure-theoretic entropy than topological entropy
For finite-state shift spaces $(X,\sigma)$, we have the variational principle:
$$
h_\text{top}(X)=\sup\{h_\mu(\sigma):\mu\text{ is a $\sigma$-invariant ergodic probability measure}\}.
$$
From what I ...
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4
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0
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162
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Recurrence under $T$ but not $T^2$ along a sequence in minimal system
Let $(X,T)$ be a non-periodic, minimal, expansive system (in particular, I am thinking of $X\subset A^{\mathbb{Z}}$ for some finite alphabet $A$).
Is it true (even known) that for any $x\in X$, there ...
- 193
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3
answers
379
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Where is the application of Anosov's shadowing lemma other than structural stability?
A few weeks ago, I learned Anosov's shadowing lemma and the application to structural stability of hyperbolic map. Despite the statement is amazing, there seems few applications of it.
Are there any ...
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0
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82
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How to find perturbation to a function defined by a perturbed implicit relation
Statement:
Suppose we have a relation $F(x,y,z)=0$ from which we can explicit find a function $z=f(x,y)$. Now suppose we have a new (perturbed) relation
$$F(x,y,z)+hG(x,y,z)=0$$
where $F$ is a known ...
- 562
11
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1
answer
396
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Determine whether the random dynamical system $f(z)=1/(U-z)$ is bounded or not
I am looking at the random iteration in $\mathbb{C}$
$$z_{n+1}=\frac{1}{U_n-z_n},\qquad n\ge 0,$$
where $(U_n)$ is an random i.i.d. sequence taking the two complex values $4+\mathrm{i}$ and $4-\mathrm{...
- 311
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0
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90
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Generalization of Stuner’s 2-color graph switching theorem to $n$-color cyclic schemes
In 1989, Stuner proved the following result:
Given a finite simple graph $G = (V, E)$, suppose each vertex $v \in V$ is initially colored white. Define an operation where, upon selecting a vertex $v \...
