Caristis fixed point theorem and strong systems of arithmetic David - - PowerPoint PPT Presentation

Caristi’s fixed point theorem and strong systems of arithmetic

David Fern´ andez-Duque Mathematics Department, Ghent University David.FernandezDuque@UGent.be Joint with Paul Shafer, Henry Towsner, and Keita Yokoyama. Wormshop 2017 Moscow, Russia

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 1 / 16

Fixed point theorems in analysis

Theorem (Banach, 1922)

Let X be a complete metric space and f : X → X be a contraction; that is, there is ρ < 1 such that d(f(x), f(y)) < ρ · d(x, y) for all x, y ∈ X. Then, there is x∗ ∈ X such that f(x∗) = x∗.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 2 / 16

Fixed point theorems in analysis

Theorem (Banach, 1922)

Let X be a complete metric space and f : X → X be a contraction; that is, there is ρ < 1 such that d(f(x), f(y)) < ρ · d(x, y) for all x, y ∈ X. Then, there is x∗ ∈ X such that f(x∗) = x∗.

Theorem (Brouwer, 1910)

Let D be a disk in Rn and f : D → D be continuous. Then, there is x∗ ∈ X such that f(x∗) = x∗.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 2 / 16

Caristi systems and Caristi’s fixed point theorem

Definition

A Caristi system is a triple (X, V, f), where

• X is a complete separable metric space,
• V : X → R≥0 is a lower semi-continuous function, and
• f : X → X is an arbitrary function,

such that ∀x∈X

• d(x, f(x)) ≤ V (x) − V (f(x))
• .

Theorem (Caristi, 1976)

If (X, V, f) is a Caristi system, then f has a fixed point.

• Henceforth, a metric space is a complete separable metric space.
• We call the V in a Caristi system a potential.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 3 / 16

Proofs of Caristi’s theorem

1 Caristi’s proof (simplified by Chi Song Wong)

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 4 / 16

Proofs of Caristi’s theorem

1 Caristi’s proof (simplified by Chi Song Wong) 2 Proof by EVP.

Theorem (Ekeland, 1974)

Every lower semi-continuous function V : X → R≥0 has a critical point, i.e. a point x∗ ∈ X such that ∀y ∈ X

• d(x∗, y) ≤ V (x∗) − V (y) → y = x∗
• David Fern´

andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 4 / 16

The Big Five subsystems of second-order arithmetic

• Two sorts: natural numbers, sets of naturals.
• Real numbers, infinite trees, etc. can all be coded in SOA.
• All systems have Σ0

1-induction, elementary arithmetical axioms.

RCA0 Computable sets exist (∆0

1 comprehension)

WKL0 RCA0 + “every infinite binary tree has an infinite path” ACA0 {n ∈ N : ϕ(n)} exists, ϕ arithmetical ATR0 Transfinite recursion for arithmetical formulas. Π1

1-CA0 {n ∈ N : ∀X⊆N ϕ(n, X)} exists, ϕ arithmetical

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 5 / 16

The strength of continuous Caristi’s theorem

Theorem (F-D S T Y)

The following are equivalent over RCA0.

1 ACA0. 2 Caristi’s theorem for continuous functions. 3 Caristi’s theorem for continuous potentials and continuous functions.

• Proof. (1 → 2). By EVP.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 6 / 16

The strength of continuous Caristi’s theorem

Theorem (F-D S T Y)

The following are equivalent over RCA0.

1 ACA0. 2 Caristi’s theorem for continuous functions. 3 Caristi’s theorem for continuous potentials and continuous functions.

• Proof. (1 → 2). By EVP.

(3 → 1). Use the fact that ACA0 is equivalent to the statement that every decreasing sequence of positive reals has an infimium.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 6 / 16

Compactness and Caristi’s theorem

Theorem (F-D S T Y)

For compact metric spaces X:

• Caristi for l.s.c. V and continuous f is equivalent to WKL0.
• Caristi for continuous V and continuous f is equivalent to WKL0.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 7 / 16

Compactness and Caristi’s theorem

Theorem (F-D S T Y)

For compact metric spaces X:

• Caristi for l.s.c. V and continuous f is equivalent to WKL0.
• Caristi for continuous V and continuous f is equivalent to WKL0.

Baire functions:

• Continuous functions are Baire class 0
• ω-limits of Baire class < ξ functions are Baire class ξ.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 7 / 16

Compactness and Caristi’s theorem

Theorem (F-D S T Y)

For compact metric spaces X:

• Caristi for l.s.c. V and continuous f is equivalent to WKL0.
• Caristi for continuous V and continuous f is equivalent to WKL0.

Baire functions:

• Continuous functions are Baire class 0
• ω-limits of Baire class < ξ functions are Baire class ξ.

Theorem (F-D S T Y)

For compact metric spaces X:

• Caristi for l.s.c. V and Baire class 1 f is equivalent to ACA0.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 7 / 16

Caristi vs. ATR0

Theorem (F-D S T Y)

Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR0.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16

Caristi vs. ATR0

Theorem (F-D S T Y)

Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR0. Facts:

1 ϕ-TR := ∀≺

• WO(≺) → ∃Z ∀ξ ∀n
• n ∈ Zξ ↔ ϕ(n, Z≺ξ)
• Zξ = {m ∈ N : m, ξ ∈ Z},
• Z≺ξ = {m ∈ N : ∃ζ≺ξ m, ζ ∈ Z}.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16

Caristi vs. ATR0

Theorem (F-D S T Y)

Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR0. Facts:

1 ϕ-TR := ∀≺

• WO(≺) → ∃Z ∀ξ ∀n
• n ∈ Zξ ↔ ϕ(n, Z≺ξ)
• Zξ = {m ∈ N : m, ξ ∈ Z},
• Z≺ξ = {m ∈ N : ∃ζ≺ξ m, ζ ∈ Z}.

2 ATR0 ≡ RCA0 + Σ0 1-TR.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16

Caristi vs. ATR0

Theorem (F-D S T Y)

Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR0. Facts:

1 ϕ-TR := ∀≺

• WO(≺) → ∃Z ∀ξ ∀n
• n ∈ Zξ ↔ ϕ(n, Z≺ξ)
• Zξ = {m ∈ N : m, ξ ∈ Z},
• Z≺ξ = {m ∈ N : ∃ζ≺ξ m, ζ ∈ Z}.

2 ATR0 ≡ RCA0 + Σ0 1-TR. 3 If ϕ is Σ0 1, there is a sequence of trees (Tξ)ξ∈N such that any path g

through Tξ codes Zξ.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16

Caristi for Baire class 1 F implies ATR0.

Proof idea. Fix (Ti)i<ω. We define a Caristi system (X, F, V ) as follows.

• X be the set of sequences of paths (gi)i<ω.
• F(

g) = lim

n→ω Fn, where Fn(

g) replaces gξ by an ‘n-approximation’ of the path through Tξ, whenever:

1 gξ(n) ∈ Tξ, and 2 ξ < n is the ≺-minimum satisfying 1.

• V (

g) = {2−n : gn is not a path through Ti}. This defines a Caristi system, whose fixed point (g∗

i )i<ω such that each g∗ i

is a path through Ti.

• David Fern´

andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 9 / 16

Leftmost paths

Theorem (Marcone)

Π1

1-CA0 is equivalent to the statement “every ill-founded tree T ⊆ N<N

has a leftmost path.”

Definition (Towsner)

• The transfinite leftmost path principle states that if T ⊆ N<N is

ill-founded and α is a well-order, then there is a path f∗ through T such that no path through T is both ΣT⊕f∗

α

and to the left of f∗.

• TLPP0 is RCA0 plus the transfinite leftmost path principle.

TLPP0 is strictly between ATR0 and Π1

1-CA0.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 10 / 16

Caristi’s fixed point theorem for Baire functions

Theorem (F-D S T Y)

Caristi for Baire functions f : X → X (with arbitrary X and l.s.c. V ) is equivalent to TLPP0. Thus in the general case:

• Caristi is equivalent to TLPP0
• Ekeland is equivalent to Π1

1-CA0

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 11 / 16

Strength of Caristi’s proof

Recall: Caristi’s proof relies on uncountable Caristi sequences.

Definition

Fix a Caristi system (X, V, f). A Caristi sequence (from x0) is a well-order (L, ≺) and a sequence (xℓ : ℓ ∈ L) ⊆ X such that      xmin L = x0 xS(ℓ) = f(xℓ) xℓ = limk<ℓ xk (ℓ ∈ Lim)

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 12 / 16

Maximal sequences

Call a Caristi sequence proper if the xℓ’s are all distinct.

Lemma (Maximal sequence principle)

Given a Caristi system (X, V, f) with f arithmetical and x0 ∈ X, there is a proper Caristi sequence with no strict, proper extensions.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 13 / 16

Maximal sequences

Call a Caristi sequence proper if the xℓ’s are all distinct.

Lemma (Maximal sequence principle)

Given a Caristi system (X, V, f) with f arithmetical and x0 ∈ X, there is a proper Caristi sequence with no strict, proper extensions. Proof of Caristi’s theorem by MSP: Any maximal sequence must have a last element, which is a fixed point of f.

• David Fern´

andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 13 / 16

The closed orbit principle

Closed orbit: If (X, f) is a dynamical system and x ∈ X, the closed orbit

• f x is the least topologically closed, f-closed set O∗ such that x ∈ O∗.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 14 / 16

The closed orbit principle

Closed orbit: If (X, f) is a dynamical system and x ∈ X, the closed orbit

• f x is the least topologically closed, f-closed set O∗ such that x ∈ O∗.

Theorem (Banach fixed point theorem 2.0)

If X is a metric space and f : X → X is a contraction, then for any x ∈ X, the closed orbit of X has a unique fixed point.

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 14 / 16

The closed orbit principle

Closed orbit: If (X, f) is a dynamical system and x ∈ X, the closed orbit

• f x is the least topologically closed, f-closed set O∗ such that x ∈ O∗.

Theorem (Banach fixed point theorem 2.0)

If X is a metric space and f : X → X is a contraction, then for any x ∈ X, the closed orbit of X has a unique fixed point.

Lemma (Closed orbit principle)

For every Caristi system (X, V, f) with f arithmetical and every x0 ∈ X, there is a ⊆-least closed set O∗ such that x0 ∈ O∗ and (∀x ∈ X)(x ∈ O∗ → f(x) ∈ O∗).

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 14 / 16

Inflationary fixed points

Definition

The arithmetical inflationary fixed point scheme is the scheme stating that if F : 2N → 2N is arithmetical and ∀X (X ⊆ F(X)), then there is a w.o. (L, ≺) with max and sets (Xα : α ∈ L) such that            Xmin L = ∅ XS(α) = F(Xα); Xγ =

α≺γ Xα

(γ ∈ Lim) F(Xmax L) = Xmax L. Stronger than Π1

1-CA0: does not require X ⊆ Y → F(X) ⊆ F(Y )

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 15 / 16

Last slide!

Theorem (F-D S T Y)

The arithmetical inflationary fixed point scheme ≡ the maximal sequence principle ≡ the closed orbit principle

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Last slide!

Theorem (F-D S T Y)

The arithmetical inflationary fixed point scheme ≡ the maximal sequence principle ≡ the closed orbit principle X f V Caristi Ekeland compact continuous continuous WKL0 WKL0 l.s.c. ACA0 Baire 1 ACA0 arbitrary continuous continuous ACA0 ACA0 l.s.c. Π1

1-CA0

Baire TLPP0

FIN

David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 16 / 16