GOEDEL DIFFEOMORPHISMS Matt Foreman September 24, 2019 CIRM, Luminy
A CLASSICAL EARLY 2O TH CENTURY QUESTION Can you tell the difference between “time running forwards” and “time running backwards”?
MATHEMATICALLY Let M be a compact smooth manifold and let φ : R × M → M be a dynamical system (say solving some ODE).
MATHEMATICALLY Let M be a compact smooth manifold and let φ : R × M → M be a dynamical system (say solving some ODE). Since R is commutative we can define : R × M → M by setting ( t, ~ x ) = � ( − t, ~ x ) and get another dynamical system with “Time running backwards.”
Is φ ∼ = ψ ?
DOES YOUR BEST PHYSICAL THEORY PROVE THAT TIME RUNS FORWARDS? HOW MANY BAD SCIENCE FICTION BOOKS ABOUT TIME TRAVEL ARE THERE?
WHAT DOES ISOMORPHISM MEAN? Since M is a compact manifold it carries a smooth volume form λ that is absolutely continuous with respect to Lebesgue measure. Is there an invertible measure preserving transformation θ that conjugates φ to ψ : θ − 1 φθ = ψ ?
BY THE ERGODIC THEOREM, MEASURE PRESERVING TRANSFORMATIONS PRESERVE STATISTICAL MEASUREMENTS.
Z VS. R If we let T : M → M be defined by T = φ (1), then we get a Z -action where the forward vs. backwards question is whether = T − 1 . T ∼
Z VS. R If we let T : M → M be defined by T = φ (1), then we get a Z -action where the forward vs. backwards question is whether = T − 1 . T ∼ We can go back to an R action from a Z -action by interpolating using the method of suspensions . So everything I say applies to R -actions.
A QUESTION OF VON NEUMANN Let ( X, B , µ ) be a standard measure space. Is there any invertible measure preserving transformation where T 6⇠ = T − 1 ? Forget about smoothness!
FIRST EXAMPLE It was not until 1951 that Anzai gave an = T − 1 by inventing the method of example of a T 6⇠ skew-product.
FIRST EXAMPLE It was not until 1951 that Anzai gave an = T − 1 by inventing the method of example of a T 6⇠ skew-product. Halmos in Math Review MR0047742 says of Anzai’s paper: “ By constructing an example of the type described in the title, the author solves (negatively) a problem proposed by the reviewer and von Neumann ”
THIS TALK IS GOING TO EXPLAIN WHY THIS IS A HARD PROBLEM
Theorem 1 (Main Theorem) There is a com- putable function F : { Codes for Π 0 1 -sentences } ! { Codes for computable di ff eomorphisms of T 2 } such that: 1. m is the code for a true statement if and only if F ( m ) is the code for a computable T , where T is measure theoretically isomorphic to T − 1 ; and 2. For m 6 = n , F ( m ) is not isomorphic to F ( n ) . The di ff eomorphisms in the range of F are ergodic.
Theorem 1 (Main Theorem) There is a com- putable function F : { Codes for Π 0 1 -sentences } ! { Codes for computable di ff eomorphisms of T 2 } such that: 1. m is the code for a true statement if and only if F ( m ) is the code for a computable T , where T is measure theoretically isomorphic to T − 1 ; and 2. For m 6 = n , F ( m ) is not isomorphic to F ( n ) . The di ff eomorphisms in the range of F are ergodic. To appear in a joint paper with J. Gaebler
FOR THOSE OF YOU WHO FORGOT YOUR FIRST YEAR LOGIC COURSE A sentence φ in the language L PA = { + , ∗ , 0 , 1 , < } is Π 0 1 if it can be written in the form ( ∀ x 0 )( ∀ x 1 ) . . . ( ∀ x n ) ψ , where ψ is a Boolean combination of equalities and inequalities of polynomials in the variables x 0 , . . . x n and the constants 0 , 1. These sentences have Goedel numbers: “codes”
WHAT IS AN (EFFECTIVELY) COMPUTABLE DIFFEOMORPHISM? We can code a modulus of continuity for a uni- formly continuous function f : T 2 → T 2 by a g : N → N such that: • To know f ( x, y ) up to n -digits it su ffi ces to supply me with the first g ( n ) digits of ( x, y ). • the computation of the digits of f ( x, y ) is recursive. A di ff eomorphism is computably C ∞ if all of its di ff erentials are computably continuous.
WHAT IS AN (EFFECTIVELY) COMPUTABLE DIFFEOMORPHISM? A function T : T 2 → T 2 is said to be a computable di ff eomorphism if there exist computable functions d : N × N → N and f : N × ( { 0 , 1 } × { 0 , 1 } ) < N → N such that d ( k, − ) and f ( k, − ) are the modulus of continuity and approximation of the k -th di ff eren- tial of T , respectively.
Computable functions of this form are also coded by Goedel numbers.
Computable functions of this form are also coded by Goedel numbers. Let’s try to see what the theorem is saying?
Theorem 1 (Main Theorem) There is a com- putable function F : { Codes for Π 0 1 -sentences } ! { Codes for computable di ff eomorphisms of T 2 } such that: 1. m is the code for a true statement if and only if F ( m ) is the code for a computable T , where T is measure theoretically isomorphic to T − 1 ; and 2. For m 6 = n , F ( m ) is not isomorphic to F ( n ) . The di ff eomorphisms in the range of F are ergodic.
Theorem 1 (Main Theorem) There is a com- putable function F : { Codes for Π 0 1 -sentences } ! { Codes for computable di ff eomorphisms of T 2 } such that: 1. m is the code for a true statement if and only if F ( m ) is the code for a computable T , where Time forwards T is measure theoretically isomorphic to T − 1 ; and backwards and 2. For m 6 = n , F ( m ) is not isomorphic to F ( n ) . The di ff eomorphisms in the range of F are ergodic.
What’s your favorite Π 0 1 statement?
CLASSICAL PROBLEMS Some examples of Π 0 1 -statements: • Riemann’s hypothesis • Goldbach’s Conjecture
BY THE THEOREM There are measures preserving transformations: • T RH such that Riemann Hypothesis if and only if T RH ⇠ = T − 1 RH . • T GB such that Goldbach’s Conjecture if and only if T GB ⇠ = T − 1 GB . Moreover T RH 6⇠ = T GB .
INDEPENDENCE RESULTS 1. “ZFC is consistent” 2. “ZFC + there is a supercompact cardinal” is consistent = T − 1 The question of whether T φ ∼ is (presumably) φ independent of ZFC.
HOW TO CHEAT: Take two di ff eomorphisms of the torus, S 0 and S 1 with S 0 ⇠ and S 1 6⇠ = S − 1 = S − 1 1 . 0 The choosing the right i , T = S i works for the Riemann Hypothesis.
BUT WE DIDN’T CHEAT. Take two di ff eomorphisms of the torus, S 0 and S 1 = S − 1 and T ∼ = T − 1 . with S ∼ The choosing the right i , T = S i works for the Riemann Hypothesis. The same i doesn’t work for all examples! Let’s look at the statement of the theorem again.
Theorem 1 (Main Theorem) There is a com- putable function F : { Codes for Π 0 1 -sentences } ! { Codes for computable di ff eomorphisms of T 2 } such that: 1. m is the code for a true statement if and only Time forwards if F ( m ) is the code for a computable T , where T is measure theoretically isomorphic to T − 1 ; and backwards and The diffeo’s faithfully 2. For m 6 = n , F ( m ) is not isomorphic to F ( n ) . code the statements The di ff eomorphisms in the range of F are ergodic.
Theorem 1 (Main Theorem) There is a com- Primitive Recursive putable function F : { Codes for Π 0 1 -sentences } ! { Codes for computable di ff eomorphisms of T 2 } such that: 1. m is the code for a true statement if and only Time forwards if F ( m ) is the code for a computable T , where T is measure theoretically isomorphic to T − 1 ; and backwards and The diffeo’s faithfully 2. For m 6 = n , F ( m ) is not isomorphic to F ( n ) . code the statements The di ff eomorphisms in the range of F are ergodic.
REVERSE MATH <= ACA_0
SOLUTION TO HILBERT’S 10TH PROBLEM (DAVIS, MATIYASEVICH, PUTNAM, ROBINSON) One phrasing of the solution is that there is a prim- itive recursive function F : { Π 0 1 − statements } → Diophantine Polynomials such that φ is true i ff F ( φ ) does not have an integer solution.
SOLUTION TO HILBERT’S 10TH PROBLEM (DAVIS, MATIYASEVICH, PUTNAM, ROBINSON) One phrasing of the solution is that there is a prim- itive recursive function F : { Π 0 1 − statements } → Diophantine Polynomials such that φ is true i ff F ( φ ) has an integer solution. Today’s theorem is an analogue for a different classical early 20th century problem.
WHAT ABOUT THE PROOF?
IN ENGLISH (SORT OF)
The proof is an adaptation of a previous result of Benjy Weiss and I: Theorem 1 In the space of C ∞ measure preserv- ing di ff eomorphisms: = T − 1 } { T : T ∼ is complete Σ 1 1 . Corollary 2 { ( S, T ) : S, T ergodic MP di ff eos and S ∼ = T } is not Borel.
This impossibility result answered another ques- tion asked by von Neumann in a 1931 paper. He proposed classifying the “statistical behavior” of smooth systems. Our result shows that this is not possible. At least with countable resources
IN PROVING THAT THEOREM We built a continuous reduction F from the space TREES to Di ff ∞ ( T 2 , λ ) such that • T is ill-founded i ff = F ( T ) − 1 • F ( T ) ∼
How do you adapt this to Π 0 1 ? Given a Π 0 1 statement ∀ n ψ you check: ψ (0) , ψ (1) , ψ (2) . . . ψ ( n ) . . . You either hit a counterexample Ω or you don’t.
1. As long you don’t hit a counterexample you = T − 1 keep trying to make T ∼ 2. If you do hit a counterexample you start tak- ing countermeasures.
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