Questions tagged [fixed-point-theorems]

A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.

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Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...
Qiaochu Yuan's user avatar
71 votes
3 answers
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Does iterating the derivative infinitely many times give a smooth function whenever it converges?

I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...
Paul Cusson's user avatar
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48 votes
4 answers
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Is Schauder's conjecture resolved?

Schauder's conjecture: "Every continuous function, from a nonempty compact and convex set in a (Hausdorff) topological vector space into itself, has a fixed point." [Problem 54 in The ...
57319's user avatar
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47 votes
6 answers
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Can we actually find any fixed points with Brouwer's theorem?

Background At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
Vidit Nanda's user avatar
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44 votes
35 answers
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Fixed point theorems

It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and ...
44 votes
2 answers
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Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...
Martin Brandenburg's user avatar
38 votes
3 answers
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Do finite groups acting on a ball have a fixed point?

Suppose that $G$ is a finite group, acting via homeomorphisms on $B^n$, the closed $n$-dimensional ball. Does $G$ have a fixed point? A fixed point for $G$ is a point $p \in B^n$ where for all $g \...
Sam Nead's user avatar
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Does the Brouwer fixed point theorem admit a constructive proof?

Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...
coudy's user avatar
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31 votes
3 answers
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Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
Mostafa Mirabi's user avatar
25 votes
5 answers
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Sperner Lemma Applications

I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of ...
24 votes
2 answers
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Periodic orbit property

A topological space $X$ satisfies the "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $...
Ali Taghavi's user avatar
22 votes
4 answers
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fixed point property for maps of compacts

Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point. Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...
Misha's user avatar
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18 votes
1 answer
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Does every automorphism of a separably rationally connected variety have a fixed point?

Let $k$ be an algebraically closed field. Let $X$ be a smooth, projective variety over $k$ that is separably rationally connected, i.e., there exists a $k$-morphism $u:\mathbb{P}^1_k \to X$ such that ...
Jason Starr's user avatar
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17 votes
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What are the major differences between real and complex Banach space?

Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa. ...
Ice sea's user avatar
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3 answers
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Banach and Knaster-Tarski fixed point theorems -- are they related?

It there any known way of obtaining the Banach fixed-point theorem from the Tarski fixed-point theorem or vice-versa?
Jacques Carette's user avatar
17 votes
1 answer
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How is this fixed point theorem related to the axiom of choice?

I'm hoping the answer to this is well-known. Let $X$ be an ordered set (i.e. poset). An inflationary operator $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) \...
Tom Leinster's user avatar
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16 votes
3 answers
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Has the Fundamental Theorem of Algebra been proved using just fixed point theory?

Question: Is there already in the literature a proof of the fundamental theorem of algebra as a consequence of Brouwer's fixed point theorem? N.B. The original post contained superfluous ...
David Bradway's user avatar
15 votes
5 answers
5k views

Converse to Banach's fixed point theorem?

Let $(X,d)$ be a metric space. Banach's fixed point theorem states that if $X$ is complete, then every contraction map $f:X\to X$ has a unique fixed point. A contraction map is a continuous map for ...
Xandi Tuni's user avatar
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14 votes
2 answers
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Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
rgvalenciaalbornoz's user avatar
14 votes
3 answers
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Topological Content of the Kakutani Fixed Point Theorem

Reading about general equilibria, the Kakutani fixed point theorem seems to be a central tool. It states (following Wikipedia) For $S \subset \mathbb{R}^n$, non-empty, compact and convex, and $\phi : ...
Aaron Bergman's user avatar
14 votes
0 answers
625 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
Gil Kalai's user avatar
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13 votes
3 answers
963 views

Closed manifolds with the fixed point property

The real projective plane ${\bf P}^2({\bf R})$ is the only closed surface with the fixed point property: all continuous maps from the plane to itself has a fixed point. The projective spaces ${\bf P}^{...
coudy's user avatar
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13 votes
0 answers
302 views

Converse of Knaster-Tarski's theorem as choice principle

Knaster-Tarski's theorem States that if $(A,\le)$ is a complete lattice, then every monotone function $f :A \to A$ has a fixed point. The proof is carried out in $\mathsf{ZF}$. By $\mathsf{KTC}$ we ...
Average-user's user avatar
12 votes
3 answers
820 views

Fixed point theorem for the uncountable power of an interval

Does the Brouwer fixed point theorem holds for the uncountable power $[0,1]^\kappa$ of the interval, $\kappa\geq\aleph_1$ ? That is, does every continuous endomorphism $[0,1]^\kappa\to [0,1]^\kappa$ ...
user494312's user avatar
12 votes
6 answers
3k views

Reference request: an elementary proof of Brouwer fixed-point theorem.

One of the elementary way to prove of the Brouwer fixed-point theorem is, making it follow from the (smooth) Non-Retraction theorem. The latter is then proven by contradiction by means of a simple ...
Pietro Majer's user avatar
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12 votes
2 answers
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A variation of the Ryll-Nardzewski fixed point theorem

Is there a fixed-point theorem that implies the following result? Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...
Anton Petrunin's user avatar
12 votes
2 answers
2k views

Connection between codata and greatest fixed points

This is, I'm afraid, another question that MSE couldn't answer. It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, ...
Ben Millwood's user avatar
12 votes
2 answers
505 views

Homeo-Fixed point property

Edit: According to comment of Michał Kukieła I revised the question A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point. ...
Ali Taghavi's user avatar
12 votes
1 answer
886 views

Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := \...
James Propp's user avatar
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12 votes
1 answer
929 views

Does a self map from the wedge sum of two spheres have either a fixed point or a point of period 2?

Let $X$ be the wedge sum of two $2$-dimensional spheres and $f$ a continuous function from $X$ into $X$. Does $f$ have either a fixed point or a periodic point of order 2? Thanks
Pedro Perez's user avatar
11 votes
4 answers
1k views

Is there a least-fixed-point formulation of inaccessible cardinals?

The infinity axiom can be formulated by defining a function $S$ as $$S(N) = \{0\} \cup \{n+1\\ |\\ n \in N\}$$ (FWIW, I'm assuming the von Neumann ordinals.) The axiom is then $$\exists I . I = S(I)...
Neil Toronto's user avatar
11 votes
1 answer
501 views

Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
Guohuan Zhao's user avatar
11 votes
1 answer
288 views

Infinite vertex-transitive graph where every automorphism has a fixed vertex

This is a follow-up to the question Connected vertex-transitive graph with the fixed-point property. In particular, it is based on a comment by user bof. Let $G = (V,E)$ be a graph with $V$ infinite. ...
Sam Hopkins's user avatar
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11 votes
0 answers
199 views

Fundamental groups of reduced subgroup lattices

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
Vidit Nanda's user avatar
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10 votes
1 answer
729 views

Equivariant Riemann-Hurwitz

The Riemann-Hurwitz formula starts with a genus $g$ algebraic curve $Y$ and a ramified cover $\pi\colon X\to Y$ of degree $N$, with ramification indices $e_P$ and computes invariants of $X$, such as ...
Ben Wieland's user avatar
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10 votes
0 answers
230 views

Can we internalize topological fixed point theorems in an effective topos?

Reflective oracles are a kind of Turing oracle that give stochastic answers about the outputs of Turing machines. This works in a self-referential way, where they can answer queries about Turing ...
Sam Eisenstat's user avatar
9 votes
6 answers
3k views

Knaster Tarski theorem, example needed

http://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem Let $L$ be a complete lattice and let $f : L \to L$ be an order-preserving function. Then the set of fixed points of $f$ in $L$ is also a ...
paul424's user avatar
  • 139
9 votes
1 answer
660 views

Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?

This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT). Yanofsky [0] has demonstrated several applications of LFPT to ...
jpt4's user avatar
  • 93
9 votes
1 answer
456 views

Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space $X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...
Darman's user avatar
  • 327
9 votes
1 answer
544 views

Is $\{x_n\}$ a Cauchy sequence?

Let $(X,d)$ be a complete metric space and $f$ a mapping of $X$ into itself. Let $\{f^n(x)\}=\{x_n\}$ be the sequence of iterated transforms. Suppose $f$ satisfies that for each $\varepsilon >0$,...
Isra El's user avatar
  • 169
9 votes
2 answers
378 views

Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?

According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point. However, these fixed points cannot be chosen ...
Alex Mennen's user avatar
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9 votes
2 answers
921 views

A question on fixed point theory

I asked this question in MSE, but I did not received any answer, so I repeat it here: https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property Assume that $0<k<n-1$, ...
Ali Taghavi's user avatar
9 votes
1 answer
441 views

Infinite-dimensional hex

Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they ...
Paul Christiano's user avatar
8 votes
2 answers
865 views

Continuity of mapping sending a function to its (brouwer) fixed point

Let $f:[0,1]^n \rightarrow [0,1]^n$ be a continuous mapping. Brouwer's fixed point theorem says that $f$ has a fixed point, i.e., some $x$ such that $f(x) = x$. Suppose we have a continuous family, i....
James Kilbane's user avatar
8 votes
1 answer
2k views

What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...
Mikhail Katz's user avatar
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8 votes
2 answers
1k views

algebraic proof of Atiyah-Bott fixed point formula?

Hi, Atiyah and Bott apparently proved the following theorem: Let $X$ be a smooth projective complex variety and $L$ a line bundle on $X$. Let $f:X\to X$ be an automorphism of $(X,L)$ with finitely ...
Nicolás's user avatar
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8 votes
2 answers
474 views

A property stronger than the fixed point property

Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-...
Ali Taghavi's user avatar
8 votes
0 answers
227 views

Do commuting homeomorphisms of the $2$-disk have a common fixed point?

Problem 2.20 (attributed to Lima) in Kirby's list of unsolved problems in low-dimensional topology asks: Are there commuting homeomorphisms of the $2$-ball $B^2$ without a common fixed point? ...
Dejan Govc's user avatar
8 votes
0 answers
270 views

Fixed points and universal maps for posets

In a recent post about f.p.p. for poset products, @M.Mirabi brought back an old-standing problem about the fixed point property of the product of two arbitrary posets which already enjoy the fixed ...
Włodzimierz Holsztyński's user avatar
7 votes
2 answers
462 views

Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?

If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...
Juan Tolosa's user avatar

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