Introduction E9F5FC

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Readings

• Jorge Soto-Andrade, Francisco Varela. Self-reference and fixed-points. A discussion and an extension of Lawveres Theorem.

Sources

• Wikipedia: Fixed point theorem
• Wikipedia: Category: Fixed point theorems
• Wikipedia: Category: Fixed points

Thoughts on fixed points

• Suhrit Dey - fixed point - consciousness
• Fixed point of Mobius transformation - important for consciousness - "doing nothing" - zeroing in on that point - taking the difference between "doing something" - relative to "doing something" we are "doing nothing"

Fixed point theorems

Continuous functions

• Brouwer fixed-point theorem For any continuous function {$f$} mapping a compact convex set to itself there is a point {$x_{0}$} such that {$f(x_{0})=x_{0}$}.
• Lefschetz fixed-point theorem The Lefschetz fixed point theorem generalizes the Brouwer fixed point theorem. Counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. The Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space, which in turn is equal to the Euler characteristic χ(X).
• Lefschetz–Hopf theorem, a stronger form of the Lefschetz fixed-point theorem.
• Atiyah–Bott fixed-point theorem A general form of the Lefschetz fixed-point theorem for smooth manifolds M, generalizing the de Rham complex constructed from smooth differential forms.
• Poincaré–Birkhoff theorem Let M be a compact differentiable manifold. Let v be a vector field on M with isolated zeroes. If M has boundary, then we insist that v be pointing in the outward normal direction along the boundary. {$\sum_{i}\mathrm{index}_{x_i}(v)=\chi (M)$} The Euler characteristic of a closed surface is a purely topological concept, whereas the index of a vector field is purely analytic. Deduced from the Lefschetz–Hopf theorem.
• Kakutani fixed-point theorem Provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. Extends the Brouwer fixed point theorem to set-valued functions.
• Schauder fixed-point theorem An extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension.
• Schaefer's fixed point theorem Useful for proving existence of solutions to partial differential equations. Let {$T$} be a continuous and compact mapping of a Banach space {$X$} into itself, such that the set {$ \{x \in X : x = \lambda T x $} for some {$0 \leq \lambda \leq 1 \} $} is bounded. Then {$T$} has a fixed point. A consequence of the Schauder fixed-point theorem.
• Leray–Schauder theorem. Generalizes Schaefer's fixed point theorem.
• Tychonoff fixed-point theorem Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X has a fixed point.
• Fixed-point theorems in infinite-dimensional spaces
• Nielsen fixed-point theorem Any map f has at least N(f) fixed points. Related to homotopy and topological fixed point theory.

Group of isometries

• Ryll-Nardzewski fixed-point theorem Functional analysis. If {$E$} is a normed vector space and {$K$} is a nonempty convex subset of {$E$} that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of {$K$} has at least one fixed point.

Contraction mapping

• Banach fixed-point theorem Let {$(X,d)$} be a non-empty complete metric space with a contraction mapping {$T\colon X\to X$}. Then T admits a unique fixed-point {$x^*$} in X (i.e. {$T(x^*) = x^*$}). Furthermore, {$x^*$} can be found as follows: start with an arbitrary element {$x_0$} in X and define a sequence {$\{x_n\}$} by {$x_n = T(x_n−1)$}, then {$x_n\rightarrow x^*$}.
• Browder fixed-point theorem A refinement of the Banach fixed-point theorem for uniformly convex Banach spaces.
• Caristi fixed-point theorem Generalizes the Banach fixed point theorem for maps of a complete metric space into itself.

Simultaneous eigenvectors

• Commuting matrices are simultaneously upper triangularizable. A result of Frobenius. See: Lie-Kolchin theorem.
• Lie-Kolchin theorem A theorem in the representation theory of linear algebraic groups. If G is a connected and solvable linear algebraic group defined over an algebraically closed field and {$\rho \colon G\to GL(V)$} a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that {$\rho (G)(L)=L$}. Equivalently, V contains a nonzero vector v that is a common (simultaneous) eigenvector for all {$\rho (g),\,\,g\in G$}.
• Lie's theorem The analog for linear Lie algebras of the Lie-Kolchin theorem. If V is a finite dimensional vector space over an algebraically closed field of characteristic 0, then for any solvable Lie algebra of endomorphisms of V there is a vector that is an eigenvector for every element of the Lie algebra. Applying this result repeatedly shows that there is a basis for V such that all elements of the Lie algebra are represented by upper triangular matrices.
• Borel fixed-point theorem A theorem in algebraic geometry. If G is a connected, solvable, algebraic group acting regularly on a non-empty, complete algebraic variety V over an algebraically closed field k, then there is a G fixed-point of V. Generalizes the Lie–Kolchin theorem.

Scott continuous function

• Kleene Fixed-Point Theorem. Suppose {$(L,\sqsubseteq )$} is a directed-complete partial order (dcpo) with a least element, and let {$ f:L\to L$} be a Scott-continuous (and therefore monotone) function. Then {$f$} has a least fixed point, which is the supremum of the ascending Kleene chain of {$f$}.

Order preserving function

• Knaster–Tarski theorem Let L be a complete lattice and let f : L → L be an order-preserving function. Then the set of fixed points of f in L is also a complete lattice.

Computable functions

• Diagonal lemma Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Let F be a formula in the language with one free variable. Then there is a self-referential sentence ψ saying that ψ has the property F. The sentence ψ can also be viewed as a fixed point of the operation assigning to each formula θ the sentence F(°#(θ)), the numeral corresponding to the Goedel number.
• Rogers's fixed-point theorem. If {$F$} is a total computable function, it has a fixed point. {$\varphi _{e}\simeq \varphi _{F(e)}$}
• Kleene's second recursion theorem For any partial recursive function {$Q(x,y)$} there is an index {$p$} such that {$\varphi _{p}\simeq \lambda y.Q(p,y).$}
• Kleene's first recursion theorem 1. For any computable enumeration operator Φ there is a recursively enumerable set F such that Φ(F) = F and F is the smallest set with this property. 2. For any recursive operator Ψ there is a partial computable function φ such that Ψ(φ) = φ and φ is the smallest partial computable function with this property.

Other

• Fixed-point combinator A higher-order function {$\mathrm {fix}$} that, for any function {$f$} that has an attractive fixed point, returns a fixed point {$x$} of that function.
• Fixed-point property Every suitably well-behaved mapping from X to itself has a fixed point. Used in topology and in order theory.
• Injective metric space Every metric map on a bounded injective space has a fixed point.
• Topological degree theory Generalizes the winding number of a curve in the complex plane. Can be used to estimate the number of solutions of an equation.