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 ...

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 ...

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.

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 ...

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 ...

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-...

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-...

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 ...

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,~~...

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 ...

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 (...

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^...

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 $\mathbb{S}^1$.
...

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 ...

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 ...

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 ...

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...

Community wiki

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 ...

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 ...

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 ...

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)
...

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 ...

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 ...

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 ...

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=...

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 ...

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 ...

3
votes

### Is there half an iteration of the QR algorithm?

Look for the Toda flow; that should do exactly what you want.

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 ...

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 $r_i$ ...

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