# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

1,577 questions
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### Does singularity confinement imply a fixed pattern of irreducible factors?

Consider a rational map $f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...
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### Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...
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### A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
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### Symmetries for Julia sets of perturbations of polynomial maps

This is a naive question. Consider the Julia sets of the map $$z \mapsto z^n + \lambda / z^k$$ with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$. For example, for $n=k=3$, ...
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### Piecewise linear expanding maps

Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose ...
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### Do complex iterates of functions have any meaning?

Using a method explained in this answer it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th iterate, where $i=\sqrt{-1}$. Here ...
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### Arithmetic dynamics and dynamics on moduli spaces

The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces. In my dissertation, I have been ...
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### Lorenz attractor path-connected?

Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure. EDIT: The answer below is unsatisfactory, and possibly ...
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### Formal group law and Koenigs function conjecture?

Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). \tag{1}$$ Consider the equation $$h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$ This equation has many ...
Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges. Suppose $f: ... 9answers 6k views ### Book recommendation for ergodic theory and/or topological dynamics? Hello, I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ... 1answer 81 views ### Discrete dynamical system and bound on norm Let$z \in \mathbb R\backslash \left\{2 \right\}$then I would like to understand the following: Consider the dynamical system with$x_i \in \mathbb C^2:$x_{i} = \left(\begin{matrix} z &&... 1answer 95 views ### 6-periodic billiards trajectory in acute triangle We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards? 0answers 134 views ### Baker map-like problem Let$S =[0, 1]^2$denote the unit square in$\mathbb R^{2}$. For any subset$A$of$S$let$A^{c}$denote its complement in$S$, and$\overline{A}$its closure in$S$. Given a measurable map$g: W \...
Consider the Tent map: $f_{μ}(x)=μx$, if $x<0.5$ and $f_{μ}(x)=μ(1-x)$ if $x≥0.5$. In this page (https://en.wikipedia.org/wiki/Tent_map) it was stated that: If $μ$ is greater than $2$ the map's ...