# Tag Info

### 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 ...
• 182k

### 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 ...
• 1,194
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.
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### 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 ...
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### 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 ...
• 839
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-...
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### 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-...
• 60.6k
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 ...
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### 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,~~...
• 1,571
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 ...
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### 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 (...
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Accepted

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### 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 ...
• 5,420
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 ...
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### Is there half an iteration of the QR algorithm?

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