# 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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### How to find a function with several bifurcations of specified type

Suppose I want to construct a function $h(x,r)$ such that it exhibits a bifurcation of type $a$ at the point $(x_{1},r_{1})$ and bifurcation of type $b$ at the point $(x_{2},r_{2})$. One alogirhtm to ...
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### How many permutations of set are there with 4 fixed points?

Given the set of numbers $A = \left \{ 1, 2, 3, 4, 5, 6, 7, 8 \right \}$. How many permutations leave exactly four numbers fixed? I've considered this as a solution... but then I stumbled upon a ...
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### How do I find the fixed points of one isometry, from the fixed points of another isometry?

I have an isometry group $G$ which acts on units of a ring $R$ (not neessarily the full isometry group). I know the set $P_f$ of fixed points in $R$ of one member $f$ of $G$. Can I use the group ...
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### $(M,d)$ complete metric space and $f : M \longrightarrow M$ such that $f^p$ is a contraction. Then, $lim f^n(x) = a$, for any $x \in M$.

I try to solve this problem: "Show that if $(M,d)$ is a complete metric space and $f: M \longrightarrow M$ is a map such that exists $p \in \mathbb{N}$ for which $f^p$ is a contraction, then, $f$ ...
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### Existence of recurrence wrt an equivalence relation implies existence of recurrence wrt equality

Assume I have $f : A \to B$, $g_a : A \to A$, $g_b : B \to B$ and $R$ an equivalence relation on $B$. If I know $f(a)$ is $R$-related to $g_b (f (g_a (a)))$ for all $a\in A$, is it the case that there ...
1 vote
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### This function has a fixed point, $F^4(x)=x$. Why?

Consider two points on the unit sphere, $C_1$ and $C_2$. Let these points be close enough such that circles of radius $r$ drawn around each point intersect at two points, $N_1$ and $N_2$. The vectors ...
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### Probabilistic Kleene's recursión theorem

Kleene's recursion theorem states that for every total computable function $f$ there is an index $e$ such that $$\phi_{f(e)} = \phi_e,$$ where $\phi_n$ is a valid enumeration of the partial computable ...
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### Fixed point of a function whose image is different to its domain.

Take a unit square in $\mathbb{R}^2$ and let $\gamma \colon [0,1] \to [0,1]^2$ be a continuous path from $[a,0]$ to $[0,b]$ where $0\le a,b < 1$ are constants. So we can think of $\gamma$ as ...
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### A Problem on Compositions of Rotations [closed]

Let $A$, $B$, and $C$ be three different points such that compositions of the following three rotations $$R(C, 256^\circ) \circ R(B, 244^\circ) \circ R(A, 220\circ)$$ has a fixed point. Determine ...
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### Does a contractible locally connected continuum have an fixed point property?

I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
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### Unique fixed point in a group action

I have been doing algebra exercises recently and I stumbled across this problem that I struggled to solve. Suppose a finite group $G$ acts on a finite set $A$ so that for every nontrivial $g \in G$ ...
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### 12 people throw their hats into a pile. They randomly pick a hat later. What is expected number of people who leave with their own hat?

Here is a similar question: Question on the 'Hat check' problem But the follow-up is, what is the standard deviation of number of people who leave with their own hat? I understand that ...
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### Conjecture:A fixed point subset of real numbers are precisely closed and bounded interval.

A topological space $(X, \tau)$ is called a fixed point space iff $f\in C(X, X)$ implies $\textrm{Fix}_f(X) \neq \emptyset$ Theorem: $(X, \tau)$ FPS implies $(X, \tau)$ connected. Proof: Suppose ...