Ball Spaces – Generic Fixed Point Theorems for Contracting Functions Franz-Viktor Kuhlmann joint work with Katarzyna Kuhlmann Dresden, 18. Januar 2018 F.-V. Kuhlmann & K. Kuhlmann Ball Spaces where fx stands for f (x). metric spaces: Banach FPT, ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], topological spaces: Brouwer FPT, Schauder FPT, partially ordered sets: Bourbaki-Witt FPT, lattices: Knaster-Tarski FPT For most FPTs some sort of “completeness” property of X is needed. FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x F.-V. Kuhlmann & K. Kuhlmann Ball Spaces metric spaces: Banach FPT, ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], topological spaces: Brouwer FPT, Schauder FPT, partially ordered sets: Bourbaki-Witt FPT, lattices: Knaster-Tarski FPT For most FPTs some sort of “completeness” property of X is needed. FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x where fx stands for f (x). F.-V. Kuhlmann & K. Kuhlmann Ball Spaces ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], topological spaces: Brouwer FPT, Schauder FPT, partially ordered sets: Bourbaki-Witt FPT, lattices: Knaster-Tarski FPT For most FPTs some sort of “completeness” property of X is needed. FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x where fx stands for f (x). metric spaces: Banach FPT, F.-V. Kuhlmann & K. Kuhlmann Ball Spaces topological spaces: Brouwer FPT, Schauder FPT, partially ordered sets: Bourbaki-Witt FPT, lattices: Knaster-Tarski FPT For most FPTs some sort of “completeness” property of X is needed. FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x where fx stands for f (x). metric spaces: Banach FPT, ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], F.-V. Kuhlmann & K. Kuhlmann Ball Spaces partially ordered sets: Bourbaki-Witt FPT, lattices: Knaster-Tarski FPT For most FPTs some sort of “completeness” property of X is needed. FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x where fx stands for f (x). metric spaces: Banach FPT, ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], topological spaces: Brouwer FPT, Schauder FPT, F.-V. Kuhlmann & K. Kuhlmann Ball Spaces lattices: Knaster-Tarski FPT For most FPTs some sort of “completeness” property of X is needed. FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x where fx stands for f (x). metric spaces: Banach FPT, ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], topological spaces: Brouwer FPT, Schauder FPT, partially ordered sets: Bourbaki-Witt FPT, F.-V. Kuhlmann & K. Kuhlmann Ball Spaces For most FPTs some sort of “completeness” property of X is needed. FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x where fx stands for f (x). metric spaces: Banach FPT, ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], topological spaces: Brouwer FPT, Schauder FPT, partially ordered sets: Bourbaki-Witt FPT, lattices: Knaster-Tarski FPT F.-V. Kuhlmann & K. Kuhlmann Ball Spaces FIXED POINT THEOREMS Given a function f : X ! X, we call x 2 X a fixed point of f if fx = x where fx stands for f (x). metric spaces: Banach FPT, ultrametric spaces: FPT of Prieß-Crampe [and Ribenboim], topological spaces: Brouwer FPT, Schauder FPT, partially ordered sets: Bourbaki-Witt FPT, lattices: Knaster-Tarski FPT For most FPTs some sort of “completeness” property of X is needed. F.-V. Kuhlmann & K. Kuhlmann Ball Spaces if there is a positive real number C < 1 such that d(fx, fy) ≤ Cd(x, y) for all x, y 2 X. Theorem (Banach’s Fixed Point Theorem) Every contracting function on a complete metric space (X, d) has a unique fixed point. Note: In metric spaces, the existence of fixed points is usually proved by means of Cauchy sequences, not by means of metric balls. Banach’s Fixed Point Theorem Let (X, d) be a metric space. A function f : X ! X is said to be contracting F.-V. Kuhlmann & K. Kuhlmann Ball Spaces Theorem (Banach’s Fixed Point Theorem) Every contracting function on a complete metric space (X, d) has a unique fixed point. Note: In metric spaces, the existence of fixed points is usually proved by means of Cauchy sequences, not by means of metric balls. Banach’s Fixed Point Theorem Let (X, d) be a metric space. A function f : X ! X is said to be contracting if there is a positive real number C < 1 such that d(fx, fy) ≤ Cd(x, y) for all x, y 2 X. F.-V. Kuhlmann & K. Kuhlmann Ball Spaces Note: In metric spaces, the existence of fixed points is usually proved by means of Cauchy sequences, not by means of metric balls. Banach’s Fixed Point Theorem Let (X, d) be a metric space. A function f : X ! X is said to be contracting if there is a positive real number C < 1 such that d(fx, fy) ≤ Cd(x, y) for all x, y 2 X. Theorem (Banach’s Fixed Point Theorem) Every contracting function on a complete metric space (X, d) has a unique fixed point. F.-V. Kuhlmann & K. Kuhlmann Ball Spaces not by means of metric balls. Banach’s Fixed Point Theorem Let (X, d) be a metric space. A function f : X ! X is said to be contracting if there is a positive real number C < 1 such that d(fx, fy) ≤ Cd(x, y) for all x, y 2 X. Theorem (Banach’s Fixed Point Theorem) Every contracting function on a complete metric space (X, d) has a unique fixed point. Note: In metric spaces, the existence of fixed points is usually proved by means of Cauchy sequences, F.-V. Kuhlmann & K. Kuhlmann Ball Spaces Banach’s Fixed Point Theorem Let (X, d) be a metric space. A function f : X ! X is said to be contracting if there is a positive real number C < 1 such that d(fx, fy) ≤ Cd(x, y) for all x, y 2 X. Theorem (Banach’s Fixed Point Theorem) Every contracting function on a complete metric space (X, d) has a unique fixed point. Note: In metric spaces, the existence of fixed points is usually proved by means of Cauchy sequences, not by means of metric balls. F.-V. Kuhlmann & K. Kuhlmann Ball Spaces where G is a totally [or partially] ordered set with minimal element 0, satisfying the following conditions for all g 2 G and x, y, z 2 X: (1) u(x, y) = 0 , x = y (2) u(x, y) = u(y, x) (3) u(x, y) ≤ maxfu(x, z), u(z, y)g [(3) if u(x, y) ≤ g and u(y, z) ≤ g, then u(x, z) ≤ g] Example: Q together with the p-adic metric is an ultrametric space. More generally, every (Krull) valuation induces an ultrametric. Ultrametric spaces An ultrametric space (X, u) is a set X together with a function u : X × X ! G, F.-V. Kuhlmann & K. Kuhlmann Ball Spaces satisfying the following conditions for all g 2 G and x, y, z 2 X: (1) u(x, y) = 0 , x = y (2) u(x, y) = u(y, x) (3) u(x, y) ≤ maxfu(x, z), u(z, y)g [(3) if u(x, y) ≤ g and u(y, z) ≤ g, then u(x, z) ≤ g] Example: Q together with the p-adic metric is an ultrametric space. More generally, every (Krull) valuation induces an ultrametric. Ultrametric spaces An ultrametric space (X, u) is a set X together with a function u : X × X ! G, where G is a totally [or partially] ordered set with minimal element 0, F.-V. Kuhlmann & K. Kuhlmann Ball Spaces (1) u(x, y) = 0 , x = y (2) u(x, y) = u(y, x) (3) u(x, y) ≤ maxfu(x, z), u(z, y)g [(3) if u(x, y) ≤ g and u(y, z) ≤ g, then u(x, z) ≤ g] Example: Q together with the p-adic metric is an ultrametric space. More generally, every (Krull) valuation induces an ultrametric. Ultrametric spaces An ultrametric space (X, u) is a set X together with a function u : X × X ! G, where G is a totally [or partially] ordered set with minimal element 0, satisfying the following conditions for all g 2 G and x, y, z 2 X: F.-V. Kuhlmann & K. Kuhlmann Ball Spaces (2) u(x, y) = u(y, x) (3) u(x, y) ≤ maxfu(x, z), u(z, y)g [(3) if u(x, y) ≤ g and u(y, z) ≤ g, then u(x, z) ≤ g] Example: Q together with the p-adic metric is an ultrametric space. More generally, every (Krull) valuation induces an ultrametric. Ultrametric spaces An ultrametric space (X, u) is a set X together with a function u : X × X ! G, where G is a totally [or partially] ordered set with minimal element 0, satisfying the following conditions for all g 2 G and x, y, z 2 X: (1) u(x, y) = 0 , x = y F.-V. Kuhlmann & K. Kuhlmann Ball Spaces (3) u(x, y) ≤ maxfu(x, z), u(z, y)g [(3) if u(x, y) ≤ g and u(y, z) ≤ g, then u(x, z) ≤ g] Example: Q together with the p-adic metric is an ultrametric space.