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# Questions tagged [fixed-points]

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. A set of fixed points is sometimes called a fixed set.

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### a global fixed point [closed]

Let $G$ be a finite subgroup of $\text{Isom}(\mathbb{E}^2)$. Show that $G$ has a global fixed point $c$ element $\mathbb{E}^2$, meaning that $g(c) = c$ for every $g$ element $G$. I admit that I ...
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### Fixed point iteration and convergence and contraction mapping

Let $p\ge 2$ be an integer, $a>0$ and $f$, $g$ be two functions defined for $x>0$ by $g(x)=ax^{1-p}$, $f(x)=x+\lambda(g(x)-x)$. Let $x_{0}$ be an appropriate initial guess, close enough to the ...
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### Permutation groups without global fixed point where every element has a fixed point

I am looking for examples (or interesting statements) of finite permutation groups $G\leq \operatorname{Sym}(n)$ with the following two properties: every element $g$ of $G$ has a fixed point, i.e., ...
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### Fixed Point Iteration Limit Point [closed]

This might sound dumb. But I cannot find a plausible answer to this question: Can fixed point iteration converge to another point than the fixed point of the continuous function?
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### "Heteroclinic orbit" for non-autonomous system?

Apologies if I am using language incorrectly, but I want to consider a non-autonomous ODE for $u(x),v(x)$ that is approximately autonomous in the limits of both $x\rightarrow\pm \infty$. An example of ...
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### Reconstructing a closure operator from a set of fixed points

Let $L$ be a lattice, not necessarily complete. We define a closure operator as a function $f\colon L\to L$ which is: idempotent, $f(f(x)) = f(x)$, isotone, $x\leq y \Rightarrow f(x) \leq f(y)$, ...
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### Stability of fixed points in a 2D discrete time dynamical system

I am studying the stability of fixed points in the Henon map for a project, and I have a question regarding it. The Henon map we are studying is given by: \begin{align*} x_{n+1}&=1+y_n-\alpha x_n^...
4 votes
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### Fixed point for some continuous function.

Let $f$ be a continuous function on $[a,b]$ ( $f: [a,b]\to \mathbb R$, $a < b$) such that $\int_a^b f(x) \, dx = \frac{b^2-a^2}{2}$. How can we prove that $f$ has a fixed point in $(a,b)$ without ...
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### Prove that $x-\frac{1}{3}\sin(x)= y$ has a unique solution for any $y \in \mathbb{R}$. [duplicate]

I am trying to prove that $x-\frac{1}{3}\sin(x)= y$ has a unique solution for any $y \in \mathbb{R}$ by a fixed point method. My aim is to apply somehow the Banach Contraction Principle. I started by ...
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### Is there a closed-form expression for this iterated mean?

Here is a simple Python implementation of the arithmetic, geometric, and harmonic means of a (non-empty) list of numbers: ...
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### How to find function $G(v_i)$, such that $\mathbb{P}\big[{\psi}(G(v_i))\big]= \frac{1}{G(v_i)}$. Is this a fixed-point problem?

The problem comes from an economic and market scenario. We have a function $$\psi(v_i)=v_i-\frac{G(v_i)-F(v_i)}{f(v_i)},$$ where the random variable $v_i$ is any real number (e.g., person $i$'s money)...
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### If $p$ is a fixed point of a $C^1$ map $f:\Bbb R\to\Bbb R$, and $|f'(p)| < 1$, then $p$ is an attracting fixed point

This is Exercise $1.5.4$ in Introduction to Dynamical Systems by Michael Brin and Garrett Stuck. Let $f: \Bbb R \to \Bbb R$ be a $C^1$ map, and $p$ be a fixed point. Show that if $|f'(p)|<1$, then ...
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### Why is a repelling fixed point also an isolated fixed point?

This is Exercise $1.5.2$ in Introduction to Dynamical Systems by Michael Brin and Garrett Stuck. Let $X$ be a locally compact metric space and $f:X\to X$ be a continuous map. We say that a fixed ...
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### A problem on fixed points in complex

Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$. I was thinking ...
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### numerically solving for the fixed points of a system of nonlinear ODEs

I was looking at an excellent lecture series on Robotics by Russ Tedrake, and he discusses Linear Quadratic Control (LQR) for system of nonlinear differential equations. So as he suggests, robots are ...
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### Grimmett and Stirzaker P57 (Letter Matching) updated

https://math.stackexchange.com/posts/2854522/edit I was stuck on step 4 of the derivation below. A secretary types n different letters together with matching envelopes, she then drops the pile down ...
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### Does there exist a non-trivial power series that equals its own coefficients on the positive integers? [duplicate]

Consider the power series $p(x) = \sum_{n=0}^{\infty} a_n x^{n}$ Suppose this power series has infinite radius of convergence. ie; $limsup_{n\rightarrow\infty} (|a_n|^{1/n}) = 0$ https://en.wikipedia....
5 votes
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### Definition of attracting fixed points in dynamical systems [book: Brin and Stuck]

I am reading the book Introduction to Dynamical Systems by Brin and Stuck. Here is the definition of attracting fixed points. Let $X$ be a locally compact metric space and $f: X \to X$ a continuous ...
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### A system of polynomials has Galois group G, a subgroup of $P_n$. Why are invariant polynomials of the roots rational, can you calculate them?

I have been studying systems of equations based upon iterating a polynomial. The complete Galois group of these systems is only a subgroup of the permutation group. There are many invariant ...
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### What are the generators of polynomials symmetric under a subgroup of the permutation group

I have been studying iterated polynomials, specifically let $P(x)=x^2+c$ and consider the equation $P(P(P(x)))=x$. After dividing out the solutions of $P(x)=x$, we have six solutions, which obey the ...
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### Finding fixed points of a specific constrained nonlinear PDE

I'm looking for attracting fixed points of the following differential equation in a vector $F$ of length $n$, and $M$ is a known $n \times n$ square matrix: \begin{align} \frac{dF}{dt} = I(F), \...