7
votes
1answer
279 views

Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$ a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right). $$ The first ten terms are: $0.75$ ...
0
votes
0answers
10 views

Stability of rest points of a time-dependent sequence

Consider the following sequence: $$x_{t+1} = (x_t)^t$$ A fixed point $a$ satisfies the following: $$a = a^t$$ for all $t$ sufficiently large. $$a(1 - a^{t-1}) = 0$$ Clearly, $a=0$ and $a = 1$ ...
7
votes
3answers
336 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
0
votes
0answers
55 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
5
votes
2answers
196 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...
1
vote
0answers
22 views

Renormalization operator R

In a Robert Devaney's book ("an introduction to chaotic dynamical systems") is approached the quadratic map $F_\mu=\mu x (1-x)$. He introduce the renormalization operator R. R is a function of ...
10
votes
2answers
129 views

What are Green's almost primes?

In a general-audience talk, Ben Green explains his famous proof with Terence Tao as an application of Szemerédi's theorem, but placing the primes within a smaller set of almost-primes in which they ...
4
votes
3answers
245 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
5
votes
2answers
218 views

Is Collatz' conjecture the only stable solution of its type?

The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...
1
vote
1answer
40 views

Limiting value of iteration $x(k+1) = A x(k) + B u(k)$ for summable $u(k)$

A matrix $A$ is known to converge such that $\lim_{k\rightarrow \infty} A^k = \bar{A} \neq 0$. We have an iteration defined as $$x(k+1) = A x(k) + B u(k), \ \ k\in \mathbb{Z}_+.$$ $\{u(k), ...
1
vote
2answers
117 views

Definition of metastability

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
8
votes
1answer
263 views

The Mandelbrot Set Membership

To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions: $$ \begin{cases} z_{n+1} &= &z_n^2 + c &\text{ for }n\geq ...
9
votes
1answer
405 views

Number of limit points of a continued exponential

Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
5
votes
3answers
613 views

The golden ratio and the logistic equation

I have this bonus question in an assignment: How is the golden ratio related to the logistic equation in discrete time? The logistic equation is $x_{n+1}=rx_n(1-x_n)$. The professor suggested ...
5
votes
5answers
240 views

Periodic sequences on finite alphabet

Let $\Sigma=\{A,B,C\}$ be an alphabet, and let $\Sigma^{\mathbb{N}}$ be the set of infinite sequences on $\Sigma$ (ie $ABCBCCCBABC...$). By outside conditions, I have several subsequences that are ...
6
votes
1answer
115 views

converging to cosine by iteration

In what sense (if at all) does the iteration $x \mapsto 2x^2 - 1$ converge to $\cos 2^n x$ in the unit interval [-1,1]? One might try to plot in Mathematica: ...