Mathematics Stack Exchange
Mathematics Stack Exchange

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.

I think it can be merged to https://math.stackexchange.com/tags/fixed-point-theorems/info

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Are there non-zero real numbers equal to their "average decimal digit"?

For some values of $x \in \mathbb{R}$, we can define what I'll call the "average digit" of $x$ and denote as $\theta(x)$ via $$\theta(x) = \lim_{n \to \infty} \frac{a_1 + \dots + a_n}{n}$$ where the $a_i$ are the uniquely defined as digits in the…
Robin
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Show that a continuous function has a fixed point

Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$. I suppose this has to do with the basic definition of continuity. The…
ghshtalt
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The entropy of entropy (or how to fix an unfair die)

$\newcommand{\on}[1]{\operatorname{#1}}$ I have recently noticed this behavior: Let $\on{P}$ be a discrete probability distribution $$ \on{P} = \left\{p_{1},\ldots, p_{n} \right\}\ \mbox{where}\ p_{1} + \cdots + p_{n} = 1,\quad p_{i} > 0\ \forall\…
Pedro Paiva
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Integral Representation of the Dottie Number

I noticed that a lot of commonly-used mathematical constants that can't be expressed in closed-form can be expressed by integrals, such as $$\pi=\int_{-\infty}^\infty \frac{dx}{x^2+1}$$ and $$\frac{1}{1+\Omega}=\int_{-\infty}^\infty…
Franklin Pezzuti Dyer
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Contraction Map on Compact Normed Space has a Fixed Point

Let $K$ be a compact normed space and $f:K\rightarrow K$ such that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ has a fixed point.
Roiner Segura Cubero
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Fixed Points Set of an Isometry

I'm reading Kobayashi's "Transformation Groups In Riemannian Geometry". I'm trying to understand the proof of the following theorem: Theorem. Let $M$ be a Riemannian manifold and $K$ any set of isometries of $M$. Let $F$ be the set of points of $M$…
Sak
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Why does the fixed point theorem justify the existence of the factorial function?

I was learning about fixed point theorem in the context of programming language semantics. In the notes they have the following excerpt: Many recursive definitions in mathematics and computer science are given informally, but they are more subtle…
Charlie Parker
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Does there exist a continuous partition of the sphere into sets of cardinality 4?

Define $X^{\{n\}}:=\{A\subseteq X:|A|=n\}$, the set of subsets of cardinality $n$. If $X$ is a topological space, $X^{\{n\}}$ can be given a topology by considering it to be a quotient of $X^n$ minus the extended diagonal. Define a continuous…
Akiva Weinberger
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Function which has no fixed points

Problem: Can anyone come up with an explicit function $f \colon \mathbb R \to \mathbb R$ such that $| f(x) - f(y)| < |x-y|$ for all $x,y\in \mathbb R$ and $f$ has no fixed point? I could prove that such a function exists like a hyperpolic function…
M.Krov
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Does the topologist's sine curve have the fixed point property?

A topological space $X$ has the fixed point property if every continuous map $f:X\to X$ has a fixed point (i.e., $x\in X$ such that $f(x)=x$). Following wikipedia, I'll consider the following two variants, each a subspace of $\mathbb R^2$. Let…
PatrickR
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Expected number of numbers that stay at their place after k swaps

Consider this problem: Let's define a list of numbers {1, 2, ..., n} in the right order. Let's make a random permutation by swapping numbers randomly. Take k random pairs of different elements and swap them. For example with k = 1 I might get the…
FluidMechanics Potential Flows
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Integer series related to the functional differential equation $f'(x) = f(f(x))$

There was an interesting discussion on the functional differential equation $$f'(x) = f(f(x)) \tag{1a}$$ The essence of which was to find a solution of $(1a)$ by Taylor expansion of $f$ about a point $x=a$, i.e. $$f(x,a) = \sum_{k\ge 0}…
Dr. Wolfgang Hintze
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Is there any noncompact manifold with the fixed point property?

This question refers to manifolds without boundary. I was thinking about Brouwer´s fixed point theorem when this came to mind, and I found the name of the property ("fixed point property"), but the information I´ve seen about it treats just compact…
Saúl RM
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Are there any path-connected sets (of $\Bbb R^2$) that guarantee two or more fixed points for any continuous bijections mapping them onto themselves?

We know by Brouwer‘s fixed-point theorem that any continuous bijection mapping the closed unit circle to itself must have a fixed point. My question: are there any path-connected sets (preferably subsets of $\mathbb R^2$) that guarantee two or more…
Franklin Pezzuti Dyer
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For a continuous function $f$ satisfying $f(f(x))=x$ has exactly one fixed point

Let $f \colon [ 0, 1] \to [0, 1]$ be a continuous map such that $$ f\big( f(x) \big) = x \ \mbox{ for each } x \in [0, 1], $$ and $$ f(x) \neq x \ \mbox{ for at least one } x \in [0, 1], $$ then how to show that $f$ has exactly one fixed…
Saaqib Mahmood
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