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Andrius Kulikauskas
 m a t h 4 w i s d o m  g m a i l
 +370 607 27 665
 My work is in the Public Domain for all to share freely.
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Divisions
Investigation: Map out the kinds of fixed point theorems.
 Interpret fixed points as perspectives.
 Consider what a perspective on a perspective means in terms of fixed points.
 Read Suhrit K. Dey's thoughts on fixed points and consciousness.
 How does chaos theory (as with the Feigenbaum constants relate the formal world and the fixed point world? What is periodic and what is not? What settles down and what doesn't? And the boundary between the two, as with the Mandelbrot set?
Readings
Sources
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 fixedpoint 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 fixedpoint 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 fixedpoint theorem.
 Atiyah–Bott fixedpoint theorem A general form of the Lefschetz fixedpoint 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 fixedpoint theorem Provides sufficient conditions for a setvalued 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 setvalued functions.
 Schauder fixedpoint 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$} has a fixed point. A consequence of the Schauder fixedpoint theorem.
 Leray–Schauder theorem. Generalizes Schaefer's fixed point theorem.
 Tychonoff fixedpoint theorem Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X has a fixed point.
 Fixedpoint theorems in infinitedimensional spaces
 Nielsen fixedpoint theorem Any map f has at least N(f) fixed points. Related to homotopy and topological fixed point theory.
Group of isometries
 RyllNardzewski fixedpoint theorem Functional analysis. If {$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$} has at least one fixed point.
Contraction mapping
 Banach fixedpoint theorem Let {$(X,d)$} be a nonempty complete metric space with a contraction mapping {$T\colon X\to X$}. Then T admits a unique fixedpoint {$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^*$}.
 Browder fixedpoint theorem A refinement of the Banach fixedpoint theorem for uniformly convex Banach spaces.
 Caristi fixedpoint theorem Generalizes the Banach fixed point theorem for maps of a complete metric space into itself.
Simultaneous eigenvectors
 Commuting matrices are simultaneously upper triangularizable. A result of Frobenius. See: LieKolchin theorem.
 LieKolchin theorem A theorem in the representation theory of linear algebraic groups. If G is a connected and solvable linear algebraic group defined over an algebraically closed field and {$\rho \colon G\to GL(V)$} a representation on a nonzero finitedimensional vector space V, then there is a onedimensional linear subspace L of V such that {$\rho (G)(L)=L$}. Equivalently, V contains a nonzero vector v that is a common (simultaneous) eigenvector for all {$\rho (g),\,\,g\in G$}.
 Lie's theorem The analog for linear Lie algebras of the LieKolchin theorem. If V is a finite dimensional vector space over an algebraically closed field of characteristic 0, then for any solvable Lie algebra of endomorphisms of V there is a vector that is an eigenvector for every element of the Lie algebra. Applying this result repeatedly shows that there is a basis for V such that all elements of the Lie algebra are represented by upper triangular matrices.
 Borel fixedpoint theorem A theorem in algebraic geometry. If G is a connected, solvable, algebraic group acting regularly on a nonempty, complete algebraic variety V over an algebraically closed field k, then there is a G fixedpoint of V. Generalizes the Lie–Kolchin theorem.
Scott continuous function
 Kleene FixedPoint Theorem. Suppose {$(L,\sqsubseteq )$} is a directedcomplete partial order (dcpo) with a least element, and let {$ f:L\to L$} be a Scottcontinuous (and therefore monotone) function. Then {$f$} has a least fixed point, which is the supremum of the ascending Kleene chain of {$f$}.
Order preserving function
 Knaster–Tarski theorem Let L be a complete lattice and let f : L → L be an orderpreserving function. Then the set of fixed points of f in L is also a complete lattice.
Computable functions
 Diagonal lemma Let T be a firstorder theory in the language of arithmetic and capable of representing all computable functions. Let F be a formula in the language with one free variable. Then there is a selfreferential sentence ψ saying that ψ has the property F. The sentence ψ can also be viewed as a fixed point of the operation assigning to each formula θ the sentence F(°#(θ)), the numeral corresponding to the Goedel number.
 Rogers's fixedpoint theorem. If {$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).$}
 Kleene's first recursion theorem 1. For any computable enumeration operator Φ there is a recursively enumerable set F such that Φ(F) = F and F is the smallest set with this property. 2. For any recursive operator Ψ there is a partial computable function φ such that Ψ(φ) = φ and φ is the smallest partial computable function with this property.
Other
 Fixedpoint combinator A higherorder function {$\mathrm {fix}$} that, for any function {$f$} that has an attractive fixed point, returns a fixed point {$x$} of that function.
 Fixedpoint property Every suitably wellbehaved 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.
