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46 votes

Does Conway's game of life admit a notion of energy?

Q: Does the temporal translation symmetry of Conway's universe give rise to a conserved quantity, that we might be able to call an "energy"? As noticed in the earliest studies of Conway's ...
Carlo Beenakker's user avatar
27 votes

Does Conway's game of life admit a notion of energy?

Conserved quantities like energy are characteristic of time-reversible dynamical systems. Conway's Game of Life is a dissipative, non-time-reversible system, and thus not likely to have any ...
Ilmari Karonen's user avatar
24 votes
Accepted

Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

This is exactly the dynamics studied by V. I. Arnold, which exhibits what is known as Arnold's tongues. See this link.
Denis Serre's user avatar
  • 51.8k
18 votes

Does Conway's game of life admit a notion of energy?

I think the question of existence of additive conserved quantities for Game of Life is not within reach of known methods. As pointed out by Ilmari Karonen, if there exist no such quantities, then ...
Ville Salo's user avatar
  • 6,447
10 votes

Do these rational sequences always reach an integer?

I want to leave a few elementary comments, maybe they will be helpful. The question asks about recurrence relation $$ u_{n+1}= \lfloor u_n \rfloor (u_n − \lfloor u_n \rfloor + 1) $$ Suppose you write ...
Ben Burns's user avatar
  • 839
10 votes
Accepted

An entire function all whose forward orbits are bounded

Given an entire function $f\colon\mathbb{C}\to\mathbb{C}$, the escaping set, $I(f)$, is the set of $z\in\mathbb{C}$ such that $f^n(z)\to\infty$. Per the Wikipedia article, the escaping set of a non-...
Gabe Conant's user avatar
  • 3,214
10 votes

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Here are a few simple remarks and one warning that are too long for a comment box. I surmise that you know most of them yourself but I'll just make them in case some reader finds any of them "non-...
fedja's user avatar
  • 60.6k
8 votes
Accepted

Topological dynamical systems with only zero-entropy factors

This question is very related to the question of lowering topological entropy, introduced in ``Can one always lower topological entropy?'' by Shub and Weiss and then very nearly solved by ...
Ronnie Pavlov's user avatar
8 votes

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

This is not a solution, but some additional numerical evidence for the non-convergent behaviour of the sequence $r_n$. The original recurrence implies for the two new variables $$ b_n ~=~ 1- r_n^2,~~...
Karl Fabian's user avatar
  • 1,571
6 votes
Accepted

Run-away functions

As noted in the question's comments by Aleksei Kulikov, a necessary and sufficient condition is given by the following: Theorem 1 A real continuous function f is a runaway function iff $f(x)=x$ has no ...
Ivan Meir's user avatar
  • 4,812
6 votes

General term formulas for nonlinear recurrence sequences

Polynomial maps $f(z)$ for which there is a general formula for the $n$-th iterate are called integrable. Besides polynomials of degree $1$, there are two types of them: a) those which are conjugate (...
Alexandre Eremenko's user avatar
6 votes
Accepted

Orbits of the function f(x)=2x (mod 1)

To be precise, $orb\big(\frac{1}{2}\big)=\mathbb{Z}\big[\frac{1}{2}\big] \cap [0,1).$ Since you say $\mathbb{Z}$ rather than $\mathbb{N}$ you mean the orbit of $y$ to include the solutions $t$ of $f^...
Aaron Meyerowitz's user avatar
6 votes
Accepted

How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?

The hyperbolic dimension of $f$ is 1 and its maximal hyperbolic set is the unit circle $\mathbb{S}^1$. First we show that a hyperbolic set for $f$ must be contained in the unit circle $\mathbb{S}^1$. ...
FMB's user avatar
  • 657
6 votes
Accepted

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

OK, finally something about the measure. I'll put it into a separate answer again for readability. If the moderators frown upon such practice, they are welcome to merge in any way they find ...
fedja's user avatar
  • 60.6k
5 votes

Periodic orbit property

The following is proved by F. Brock Fuller in "The Existence of Periodic Points," Annals of Mathematics, Vol. 57, 1953, pp. 229-230: Theorem. Let X be a compact simplicial complex with Betti ...
Moe Hirsch's user avatar
5 votes
Accepted

Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?

Explicit solutions for arbitrary $r$ exist in various forms: Logistic map: an analytical solution (1995) represents the solution as a power of a transfer matrix. An explicit solution for the ...
Carlo Beenakker's user avatar
5 votes

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Just another illustration (cw) - color-coded reciprocals of distances from the origin after 14 iterations: I believe this shows that there is no easy answer...
5 votes

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

I still cannot figure out what is going on in the measure-theoretic sense, but the topological dynamics has become a bit clearer. Formally we will show that the set of points that stay sufficiently ...
fedja's user avatar
  • 60.6k
4 votes
Accepted

Showing that the inverse of a function is approximately equivalent to $\frac{1}{n^{1/\alpha}}$

The right part of $f$ being not relevant, we consider the diffeo $f:[0,1/2]\to[0,1]$, $f(x)=x+2^\alpha x^{1+\alpha}$, whose inverse map $g:[0,1]\to[1,1/2]$ is strictly increasing with unique fixed ...
Pietro Majer's user avatar
  • 57.7k
4 votes

Topological dynamical systems with only zero-entropy factors

YES under strong additional assumptions. Theorem. Let $X$ be compact metrizable and zero-dimensional, and let $T : X \to X$ be an aperiodic homeomorphism. If $(X, T)$ has only zero-entropy proper ...
Ville Salo's user avatar
  • 6,447
4 votes
Accepted

State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

This question was answered recently by Geschke, Grebík, and Miller: S. Geschke, J. Grebík, and B. D. Miller, "Scrambled Cantor sets," Proceedings of the AMS 149 (link to the arxiv version) ...
Will Brian's user avatar
  • 17.6k
4 votes
Accepted

Condition for 3×3 block matrix to be stable

$\newcommand{\al}{\alpha}\newcommand\la\lambda\newcommand\R{\mathbb R}$Such a construction of $M$ and $\al$ is always possible. Indeed, take any complex $\la$. Rearranging columns and rows of the ...
Iosif Pinelis's user avatar
3 votes
Accepted

Reversal of open cover with topologically transitive dynamical system

No, even if we assume $\nu$ to be invariant under $\phi$. Let $X = \{0,1\}^\mathbb{Z}$ be the set of two-way infinite binary sequences with the prodiscrete topology, and let $\phi$ be the left shift ...
Ilkka Törmä's user avatar
3 votes

Why do finitely many cluster variables imply finitely many y-variables?

Let $\Sigma:=(\mathbf{x},\mathbf{y},B)$ be an initial seed where $\mathbf{x}:=\{x_1,\ldots,x_n\},\mathbf{y}:=\{y_1,\ldots,y_n\}$ and let $\mathbf{c} := \{x_{n+1},\ldots,x_m\}$ denote the set of frozen ...
Jon Wilson's user avatar
3 votes

General term formulas for nonlinear recurrence sequences

See Mathworld, where it is stated, "While some quadratic maps are solvable in closed form, most are not." Examples with a closed-form solution are $p_n=p_{n-1}^2$, $p_n=p_{n-1}^2+1$ with $p_0=1$, $p_n=...
Gerry Myerson's user avatar
3 votes

Random reflections unexpectedly produce banded distributions

Let $p_i$ be at distance $r_i > 2$ from the origin, WLOG assume $p_i = (r_i, 0)$. Let $q_i = (\cos \alpha, \sin \alpha)$, and $v_i = (\cos \beta, \sin \beta)$ be a direction vector of $M_i$. By ...
Mikhail Tikhomirov's user avatar
3 votes
Accepted

Invariant distributions for iterated random variables (stochastic dynamical systems)

To have here the invariant distribution with cdf $F$ given by $F(x)=x^2$ for $x\in[0,1]$, all that is needed is a change of variables. More generally, let $F$ be the cdf of any non-atomic distribution ...
Iosif Pinelis's user avatar
3 votes

Is there half an iteration of the QR algorithm?

Look for the Toda flow; that should do exactly what you want.
Federico Poloni's user avatar
3 votes

Does every proximal dynamical system have zero topological entropy?

Maybe there is an easier example, but here is an example of a proximal system with positive entropy. The dynamical system is a so-called subshift $(X, \sigma)$, where $X$ is a closed shift-invariant ...
Ronnie Pavlov's user avatar
2 votes

Random reflections unexpectedly produce banded distributions

As said by Sangchui Lee, we are interested in $r_i = |p_i|$ and we assume that the reflections induce enougth random rotation such that the "bands" of radius $r$ just reveal the number of time $r_i$ ...
RaphaelB4's user avatar
  • 4,341

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