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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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)
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $\Gamma \backslash G$ ...
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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 $k\ge 1$, ...
