Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling F(x)=x, under some conditions on the function F. Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, ...

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What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
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651 views

Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
22
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325 views

Does every continuous map from $\mathbb{H}P^{2n+1}$ to itself have a fixed point?

This question is motivated by Fixed point property of Cayley plane and idle curiosity. In the link, it is shown that every continuous map from the Cayley plane to itself has a fixed point and the ...
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185 views

Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
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Generalization of “easy” 1-D proof of Brouwer fixed point theorem

In topology class, before we learned the homotopy-based proof of Brouwer's fixed point theorem, the professor mentioned the easy proof of the one-dimensional case: just draw the graph $f(x) = x$ in ...
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Contraction mapping in an incomplete metric space

Let us consider a contraction mapping $f$ acting on metric space $(X,~\rho)$ ($f:X\to X$ and for any $x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1$). If $X$ is complete, then there ...
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In this case, does $\{x_n\}$ converge given that $\{x_{2m}\}$ and $\{x_{2m+1}\}$ converge?

I'm playing around with a sequence $\{x_n\}$ defined by $$ x_{n+1}=\frac{\alpha+x_n}{1+x_n}=x_n+\frac{\alpha-x_n^2}{1+x_n}. $$ Here $\alpha\gt 1$, and $x_1\gt\sqrt{\alpha}$. I'm trying to compute ...
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332 views

Common knowledge as a fixed point

I read on a wikipedia page that from the modal logic formalization CK can be formulated as a fixed point. If it also holds for the set theory formalization? If it does, where I can find about it? ...
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Finding a functor satisfying a recursive equation

Say I want to find a functor satisfying something like: \[FA = 1 \sqcup (A \times FA).\] Equivalently, I want to find for each $A$ a fixed point of: \[GB = 1 \sqcup (A \times B)\] (I'll worry about ...
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Is the Knaster-Tarski Fixed Point Theorem constructive?

According to Tarski's Fixed Point Theorem, for a complete lattice $L$, and monotone function $f:L \rightarrow L$, the set of fixed points of $f$ forms complete lattice. Definition of $lfp(f)$ and ...
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Schwarz's Lemma, fixed points question

This is from an old qualifying examination question. If f is holomorphic in the unit disk $D$ and $|f(z)|<1$ for all $z\in D$. Suppose also that $f$ has two distinct fixed points in $D$ then ...
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Continuous Brouwer's fixed point theorem via Stokes's theorem?

Let $B$ denote the closed unit ball in $\mathbf{R}^n$. Brouwer's fixed point theorem states that every continuous map $f:B\to B$ has a fixed point. There is a simple proof using Stokes's theorem, at ...
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Do there exist sets $A\subseteq X$ and $B\subseteq Y$ such that $f(A)=B$ and $g(Y-B)=X-A$?

This is a little exercise I've been fiddling with for a while now. Let $f\colon X\to Y$ and $g\colon Y\to X$ be functions. I want to show that there are subsets $A\subseteq X$ and $B\subseteq Y$ ...
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Minimax Theorems V.S. Fixed Point Theorems.

Is there any relationship between the minimax theorems $$ \mbox{Regularly hypothesis on } f:U\times V\to\mathbb{R} \implies \min_{u}\max_{v}f(u,v)=\max_{v}\min_{u}f(u,v) $$ and fixed point ...
7
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Fixed point: sets and measures

Let $X$ be a Borel space with a Borel measure $\mu$. Suppose $\xi: X\times X\to\mathbb R_{\geq 0}$ is a continuous function and put $s(x) = \{y\in X:\xi(x,y) = 0\}$. For any set $b\in\mathcal B(X)$ we ...
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Fixed Point Theorems

Theorem 1. Let $B=\{x\in \mathbb R^n :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^n$ . Any continuous function $f:B\rightarrow B$ has a fixed point. Theorem 2. Let $X$ be a finite dimensional ...
7
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Least common fixed-point

I have been reading a book, "Introduction to Lattices and Order", and I'm trying to solve exercise 8.29 as the following in it: Suppose that $P$ is a complete lattice and let $F$ and $G$ be ...
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Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
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369 views

A fixed point theorem for the unit disk?

In Dynamical Systems and Ergodic Theory by Pollicott and Yuri, there is an easy, one dimensional, fixed point theorem: If $T$ is a continuous map on a closed interval $J$ so that $T(J)\supseteq ...
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Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point

I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem. I think I should use the ...
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Fixed points of the Gamma Function?

I am interested in complex values of $z$ such that $$ \Gamma (z) =z$$ Clearly, the one trivial value of $z$ is 1. Also, looking at a graph of the gamma function on the real axis, I can tell that there ...
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How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
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periodic point of homeomorphism of plane

Let $h$ be a homeomorphism of $\mathbb{R}^2$ onto itself such that $h(K)=K$ for some compact subset $K$ of $\mathbb{R}^2$. Show that $h$ has a periodic point in $\mathbb{R}^2$.
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Is there a simple proof of Borsuk-Ulam, given Brouwer?

(Brouwer) Any continuous function from a convex compact subset K of a Euclidian space to itself has a fixed point. Given this lemma, is there a simple proof of: (Borsuk-Ulam) Any continuous ...
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Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
6
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A function over the integers and its fixed points

Define $f:\mathbb{N}\rightarrow\mathbb{N}$ as follows, $f(n)$ is the number of times the digit "1" is needed if we were to write all integers between 1 and $n$ (inclusive) in base 10. So for example ...
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A fixed point theorem [duplicate]

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element of $\cal ...
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Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
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Why is convexity a requirement for Brouwer fixed points? Shouldn't “no holes” be good enough?

Brouwer's fixed point theorem: Every continuous function $f$ from a convex compact subset $K$ of a Euclidean space to $K$ itself has a fixed point. I am wondering why the word "convex" is in ...
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Banach theorem example

By Banach fixed point theorem, if a metric on a metric space $X$ is such that $d(f(x),f(y))\leq K d(x,y)$ for $K\in (0,1)$ then $f$ has one unique fixed point. Is there an example where ...
5
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How to show something is a contraction?

If we let $X$ be a complete metric space, and let $S:X\to X$ be a map, such that $S^m$ is a contraction. We now want to show, that $S$ has a unique fixed point This is what I've thought so far: Due ...
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Opposite of a contraction mapping

I am taking Real Analysis and we recently went over the Banach Fixed-point Theorem, also commonly known as the Contraction Mapping Theorem which states: If $(X,d)$ is a complete metric space, and ...
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Can the proof of fixed point theorems ever be constructive?

Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these ...
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680 views

Continuous function on unit circle has fixed point

The question I have is: Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.
5
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No fixed points imply no periodic points

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth injective function with $\operatorname{det}[f'(x)]\not=0 $ for all $x\in\mathbb{R}^n$. Moreover assume that $f$ has no fixed points. Can $f$ have a ...
5
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Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
5
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Is Poincare-Hopf index theorem connected with Leftschetz fixed point theorem?

Lefschetz Fixed Point Theorem: For a compact triangulable space $X$, and a continuous map $f:X\rightarrow X$, we have ...
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Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
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Why $e^x$ never equal $x$?

Je veux savoir pourquoi $x=e^x$ n'a aucune solution dans $\Bbb R$. Lorsque j'ai essayé de tracer le graphe de la fonction $e^x$, j'ai trouvé en fait qu'elle est une fonction strictement croissante ...
4
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267 views

Fixed points in category theory

Is there a usual way to express the concept of fixed points in category theory? What would I say to express that a morphism has a fixed point? Thank you!
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What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent ...
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banach fixed point theorem

Let $T:X \to X$ be a map on a complete non-empty metric space. Assume that for all $x$ and $y$ in $X$, $\sum_n d(T^n(x),T^n(y))<\infty$. Then $T$ has a unique fixed point. guess: I assume that the ...
4
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In a complete lattice every monotone function has a fixpoint (Knaster–Tarski Theorem)

L is a complete lattice, so every subset has a supremum and infimum. In addition, there exists a function $f:L \rightarrow L$ such that $a \leq b$ implies $f(a) \leq f(b)$. Prove that there exists ...
4
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Is there a simple way to prove the Brouwer fixed Point theorem?

The quest may be for references but I want to know if there is a simple way to prove the Brouwer fixed point theorem! That is if a function $f:\bar{B}\to\bar{B}$ is continuous then $f$ admits one ...
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Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
4
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Does a compact connected complete linear order have the fixed point property?

Would the same arguments used for showing $[0,1]$ has the fixed point property hold in this general case? What could go wrong? EDIT: The fixed point property can be interpreted in two ways that I ...
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Fix point of squaring numbers mod p

Take the set of integers $\{0, 1, .., p-1\}$, square each element, you get the (smaller) set of quadratic residues. Repeat until you get a fix point set. The size of this set is a function of $p$. ...
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generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
4
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Is there any way to give sense to a geometric/visual proof?

Suppose one is given the following visual proof that $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$$ which is the following construction over $[0,1]\times[0,1]$ What this is ...
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Fixed point property of Cayley plane

I want to know whether the Cayley plane has fixed point property or not. I think it is so but I am not able to prove this. It certainly does not admit maps of period two without fixed points. By fixed ...