Generalization of Browder's F.P. Theorem?
Hello -- Does anyone know of a generalization of Browder's Fixed Point Theorem from R^n to arbitrary topological spaces, or to categories of same? ***************** Theorem (Browder, 1960): Suppose that S is a non-empty, compact, convex subset of R^n, and let f: [0,1] x S --> S be a continuous function. Then the set of fixed points { (x,s) | s = f(x,s), x \in [0,1] and s \in S } contains a connected subset A such that the intersection of A with {0} x S is non-empty and the intersection of A with {1} x S is non-empty. ***************** Many thanks, -- Peter McBurney University of Liverpool, UK
I'm intrigued by Peter McBurney's question [below]. It looks rather like a question of the constructive content of Brouwer's fixed point theorem. Suppose S is (homeomorphic to) an n-cell. Then in the internal logic of the topos of sheaves over [0,1], f is just a continuous endomap of S. If Brouwer's theorem were constructively true then f would have a fixpoint, and that would come out as a continuous section of the projection [0,1]xS -> [0,1]. More precisely, it would be a map g: [0,1] -> S such that f(x, g(x)) = g(x) for all x. If this existed then the set A = {(x, g(x))| x in [0,1]} would be as required. However, the proof of Brouwer that I've seen is not constructive - it goes by contradiction. So maybe the requirements on A are a way of getting constructive content in Brouwer's result. What is known constructively about Brouwer's fixed point theorem? Steve Vickers. Peter McBurney wrote:
Hello --
Does anyone know of a generalization of Browder's Fixed Point Theorem from R^n to arbitrary topological spaces, or to categories of same?
*****************
Theorem (Browder, 1960): Suppose that S is a non-empty, compact, convex subset of R^n, and let
f: [0,1] x S --> S
be a continuous function. Then the set of fixed points
{ (x,s) | s = f(x,s), x \in [0,1] and s \in S }
contains a connected subset A such that the intersection of A with {0} x S is non-empty and the intersection of A with {1} x S is non-empty.
*****************
Many thanks,
-- Peter McBurney University of Liverpool, UK
On Thu, 16 Jan 2003, Steven J Vickers wrote:
I'm intrigued by Peter McBurney's question [below]. It looks rather like a question of the constructive content of Brouwer's fixed point theorem.
Suppose S is (homeomorphic to) an n-cell. Then in the internal logic of the topos of sheaves over [0,1], f is just a continuous endomap of S. If Brouwer's theorem were constructively true then f would have a fixpoint, and that would come out as a continuous section of the projection [0,1]xS -> [0,1]. More precisely, it would be a map g: [0,1] -> S such that f(x, g(x)) = g(x) for all x. If this existed then the set A = {(x, g(x))| x in [0,1]} would be as required.
However, the proof of Brouwer that I've seen is not constructive - it goes by contradiction. So maybe the requirements on A are a way of getting constructive content in Brouwer's result.
What is known constructively about Brouwer's fixed point theorem?
Steve Vickers.
Similar thoughts had occurred to me. Brouwer's theorem is clearly not constructive, since it doesn't hold (even locally) continuously in parameters (consider a path in the space of endomaps of [0,1] passing through the identity, where the fixed point `flips' from one end of the interval to the other as it does so). However, Browder's result would seem to suggest that the `locale of fixed points of f' (that is, the equalizer of f and the identity in the category of locales) might be consistent (that is, `inhabited') in general, even though it may not have any points. It's certainly conceivable that that might be true constructively, though I can't see how to prove it -- but it isn't the full content of Browder's theorem. Peter Johnstone
In Errett Bishop's Constructive Analysis (anyone who is interested in analysis over a topos absolutely must know this book), he proves that for any continuous endomorphism f of a disk and for every eps > 0, there is a point x in the disk for which |f(x) - x| < eps. A couple of points should be made. First f has to be constructible and eps has to be provably positive. For Bishop, a real number is an equivalence class of pairs of RE sequences of integers a_n and b_n such that for all m,n |a_n/b_n - a_m/b_m| < 1/m + 1/n and a function is constructive if it is a machine for turning one such sequence into another. To be continuous, it there must be a function delta(eps) that produces for each eps > 0, a delta(eps) such that |x - y| < delta(eps) implies that |f(x) - f(y)| < eps. (In fact, there is a non-constructive proof that every constructive function is continuous.) Bishop then claims, without proof as far as I can see, that there is a fixed point free endomorphism of the disk. What this means is that when you extend this function to all reals, any fixed point is not a constructible real number. On Thu, 16 Jan 2003, Steven J Vickers wrote:
I'm intrigued by Peter McBurney's question [below]. It looks rather like a question of the constructive content of Brouwer's fixed point theorem.
Suppose S is (homeomorphic to) an n-cell. Then in the internal logic of the topos of sheaves over [0,1], f is just a continuous endomap of S. If Brouwer's theorem were constructively true then f would have a fixpoint, and that would come out as a continuous section of the projection [0,1]xS -> [0,1]. More precisely, it would be a map g: [0,1] -> S such that f(x, g(x)) = g(x) for all x. If this existed then the set A = {(x, g(x))| x in [0,1]} would be as required.
However, the proof of Brouwer that I've seen is not constructive - it goes by contradiction. So maybe the requirements on A are a way of getting constructive content in Brouwer's result.
What is known constructively about Brouwer's fixed point theorem?
Steve Vickers.
Peter McBurney wrote:
Hello --
Does anyone know of a generalization of Browder's Fixed Point Theorem from R^n to arbitrary topological spaces, or to categories of same?
*****************
Theorem (Browder, 1960): Suppose that S is a non-empty, compact, convex subset of R^n, and let
f: [0,1] x S --> S
be a continuous function. Then the set of fixed points
{ (x,s) | s = f(x,s), x \in [0,1] and s \in S }
contains a connected subset A such that the intersection of A with {0} x S is non-empty and the intersection of A with {1} x S is non-empty.
*****************
Many thanks,
-- Peter McBurney University of Liverpool, UK
There seems to be some confusion about the theorem Peter McBurney asked about. The reference he cited was by Felix E. BROWDER (1960) who proved a number of fixed point results of considerable interest to functional analysts. The theorem the list seems to be discussing is due to BROUWER (Math. Ann. 69(1910) and 71(1912). Carl Futia
At 11:19 17/01/03 EST, you wrote:
There seems to be some confusion about the theorem Peter McBurney asked about.
The reference he cited was by Felix E. BROWDER (1960) who proved a number of fixed point results of considerable interest to functional analysts.
The theorem the list seems to be discussing is due to BROUWER (Math. Ann. 69(1910) and 71(1912).
Carl Futia
No, there is no confusion. Browder's theorem appears to provide insight into the constructive content of Brouwer's theorem. Both are being discussed. Steve Vickers.
participants (6)
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Carl Futia -
Michael Barr -
Peter McBurney -
Prof. Peter Johnstone -
S Vickers -
Steven J Vickers