Speedups of Z d -odometers David M. McClendon Ferris State - - PowerPoint PPT Presentation

Speedups of Zd-odometers

David M. McClendon

Ferris State University Big Rapids, MI, USA

joint with Aimee S.A. Johnson (Swarthmore)

David McClendon Speedups of Zd -odometers

Some words about Jane

David McClendon Speedups of Zd -odometers

This talk is about actions of Zd

Definition A Zd− measure-preserving system (Zd-m.p.s.) is a quadruple (X, X, µ, T) where (X, X, µ) is a Lebsegue probability space and T = {Tv : v ∈ Zd} is an action of Zd on X by measure-preserving transformations. Definition A Zd− Cantor minimal system (Zd-C.m.s.) is a pair (X, T) where X is a Cantor space and T = {Tv : v ∈ Zd} is a minimal action of Zd on X by homeomorphisms. In either situation, we can write T = (T1, ..., Td) where Tj is shorthand for the action of standard basis vector ej.

David McClendon Speedups of Zd -odometers

Speedups of actions of Zd

Definition A cone C is the intersection of Zd − {0} with any open, connected subset of R2 bounded by d distinct hyperplanes passing through the origin.

• Example in Z2:

✏✏✏✏✏✏✏✏✏✏ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡

• C

David McClendon Speedups of Zd -odometers

Speedups of actions of Zd

Definition A C−speedup of Zd−action T = (T1, ..., Td) is another Z2−action Tp = (T 1, ...T d) (defined on the same space as T) such that T j(x) = Tpj(x)(x) for some function p = (p1, ..., pd) : X → Cd. p is called the jump function or the speedup function. Remark: The p must be defined so that the T j commute (so one cannot simply speed up the generators Tj independently to obtain a speedup of T).

David McClendon Speedups of Zd -odometers

A picture to explain (d = 2)

• x

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻

T1 T2

✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✗ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✿

C

✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘ ✘ ✘✘✘✘ ✘✘ ✘ ✏✏✏✏✏✏✏✏ ✏ ✶ ✁ ✁ ✁ ✁ ✁ ✁ ✕

• ✒

✟✟✟✟✟ ✟ ✯ ✘✘✘✘✘✘✘✘✘✘✘ ✘ ✿

• ✒

T 1 T 1 T 1 T 2 T 2 T 2

• Here, Tp = (T 1, T 2) is a C−speedup of T = (T1, T2).

In particular, for the indicated point x, we have p(x) = ((3, 1), (1, 2)).

David McClendon Speedups of Zd -odometers

Why is this called a “speedup”?

When d = 1, there are two cones: C+ = {1, 2, 3, ...} and C− = {−1, −2, −3, ...}. A C+−speedup looks like this: · · ·

T

T

• T p
• T
• T p

T

• T p
• T
• T p
• T

T

• T p
• T · · ·

David McClendon Speedups of Zd -odometers

The big picture

Question Given two Zd-actions (X, T) and (Y , S), when is there a speedup

• f T isomorphic to S?

The word isomorphic means: measurably conjugate, if T and S are Zd-m.p.s. topologically conjugate, if T and S are Zd-C.m.s. Notation Write T

C S

if there is a C-speedup of T isomorphic to S.

David McClendon Speedups of Zd -odometers

The big picture

Question Given two Zd-actions (X, T) and (Y , S), when is there a speedup

• f T isomorphic to S?

The word isomorphic means: measurably conjugate, if T and S are Zd-m.p.s. topologically conjugate, if T and S are Zd-C.m.s. Notation Write T S if for any cone C ⊆ Zd, T

C S.

David McClendon Speedups of Zd -odometers

The big picture

Question Given two Zd-actions (X, T) and (Y , S), when is there a speedup

• f T isomorphic to S?

The word isomorphic means: measurably conjugate, if T and S are Zd-m.p.s. topologically conjugate, if T and S are Zd-C.m.s. Notation Write T

adjective

• S

if T S via a speedup function p which is that adjective.

David McClendon Speedups of Zd -odometers

History of speedups: ergodic theory

Theorem (Neveu 1969) Suppose (X, T) and (Y , S) are m.p.t.s. If T

integrable

• S, then

h(S) =

• p dµ
• h(T).

Theorem (Arnoux-Ornstein-Weiss 1985) Suppose (X, T) and (Y , S) are m.p.t.s, where T is ergodic. Then T mble S. Theorem (Johnson-M, 2014) Suppose (X, T) and (Y , S) are Zd-m.p.s., where T is ergodic. Then T mble S.

David McClendon Speedups of Zd -odometers

History of speedups: ergodic theory

The basic framework of the AOW (and JM) proofs can be traced to a proof of Dye’s Theorem given by Hajian, Ito and Kakutani in

• 1975. Recall:

Theorem (Dye 1963) Suppose (X, T) and (Y , S) are ergodic Zd-m.p.s. Then T and S are (measurably) orbit equivalent. Big picture idea When T and S are orbit equivalent, we think of T and S as “having the same orbits”. When T S, each T-orbit is partitioned into distinct S-orbits. This suggests that the “speedup relation” has something to do with orbit equivalence.

David McClendon Speedups of Zd -odometers

History of speedups: topological dynamics, d = 1

Theorem (Giordano-Putnam-Skau 1995) Let T and S be two Cantor minimal systems. Then TFAE:

1 T and S are toplogically orbit equivalent. 2 T and S have isomorphic dimension groups.

Theorem (Ash) Let T and S be two Cantor minimal systems. Then TFAE:

1 T lsc

S. (“lsc” is “lower semicontinuous”)

2 There is a surjective group homomorphism from the dimension

group of S to the dimension group of T, preserving the positive cones and distinguished order units of those groups.

David McClendon Speedups of Zd -odometers

History of speedups: topological dynamics, d = 1

Much more restrictive things happen when one asks that the speedup function p be continuous (hence bounded, since X is compact): Theorem (Alvin-Ash-Ormes) Let T be an odometer (more specifically, a Z-odometer), and suppose T cts

• S. If S is minimal, then S is a Z-odometer which is

topologically conjugate to T. Question What happens with continuous speedups of Zd-odometers?

David McClendon Speedups of Zd -odometers

Zd-odometers

Zd-odometers were introduced by Cortez in 2004. They are defined as follows: The phase space Let Zd ≥ G0 ≥ G1 ≥ G2 ≥ G3 ≥ · · · be a decreasing sequence of subgroups of Zd, each of which have finite index in Zd, such that

∞

∩

j=0 Gj = {0}. Let X be the inverse

limit X = lim

← − (Zd/Gj).

David McClendon Speedups of Zd -odometers

Zd-odometers

Zd-odometers were introduced by Cortez in 2004. They are defined as follows: The phase space Each element x of X is formally an infinite sequence of cosets, i.e. something like x = (x0 + G0, x1 + G1, x2 + G2, ...) where the xj are “commensurate”, i.e. since Gj ≥ Gj+1, there’s a natural map πj : Zd/Gj+1 → Zd/Gj; for such a sequence to be in X we require that, for all j, πj(xj+1 + Gj+1) = xj + Gj.

David McClendon Speedups of Zd -odometers

Zd-odometers

Zd-odometers were introduced by Cortez in 2004. They are defined as follows: The action X is a Cantor space, and also a topological group with addition defined coordinate-wise, where the addition in the jth coordinate is the usual (vector) addition in the quotient group Zd/Gj. Given any v ∈ Zd, we can “convert” v into an element of X by setting τ(v) = (v + G0, v + G1, v + G2, ...) Define the action T of Zd on X by Tv(x) = x + τ(v). (X, T) is a Zd-C.m.s. called a Zd-odometer.

David McClendon Speedups of Zd -odometers

Speedups of Zd−odometers

Theorem (Johnson-M) Let T be a Zd−odometer, and suppose T cts

• S. If S is minimal,

then S is topologically conjugate to a Zd−odometer. Same result as d = 1 (AAO), but not a similar proof.

David McClendon Speedups of Zd -odometers

Speedups of Zd−odometers

Theorem (Johnson-M) Let T be a Zd−odometer, and suppose T cts

• S. If S is minimal,

then S is topologically conjugate to a Zd−odometer. Same result as d = 1 (AAO), but not a similar proof. Theorem (Johnson-M) Let T be a Zd−odometer, and suppose T cts

• S. Even if S is

minimal, S need not be topologically conjugate to T. Not same result as d = 1.

David McClendon Speedups of Zd -odometers

Speedups of Zd−odometers

Theorem (Johnson-M) Let T be a Zd−odometer, and suppose T cts

• S. If S is minimal,

then S is topologically conjugate to a Zd−odometer. Same result as d = 1 (AAO), but not a similar proof. Theorem (Johnson-M) Let T be a Zd−odometer, and suppose T cts

• S. Even if S is

minimal, S need not be topologically conjugate to T. Not same result as d = 1. Actually... it is (kind of) the same result as d = 1, if one considers orbit equivalence.

David McClendon Speedups of Zd -odometers

Orbit equivalence and speedups of Zd-odometers

Theorem (Johnson-M) Let T and S be Zd−odometers. TFAE:

1 For some cone C ⊆ Zd, T cts

• C S.

2 T and S are continuously orbit equivalent.

Remark: When d = 1, statement (3) of the above theorem implies T and S are flip conjugate (Boyle 1983, Boyle-Tomiyama 1998), and since T and S are Z-odometers they are each isomorphic to their inverses. The AAO result follows.

David McClendon Speedups of Zd -odometers

Orbit equivalence and speedups of Zd-odometers

Theorem (Johnson-M) Let T and S be Zd−odometers. TFAE:

1 For some cone C ⊆ Zd, T cts

• C S.

2 T and S are continuously orbit equivalent.

A key concept used in the proof is that of structural conjugacy

• f odometers, recently introduced by Cortez & Medynets.

This gives an algebraic condition on sequences of subgroups defining two

• dometers which reveals whether or not they are continuously orbit

equivalent.

David McClendon Speedups of Zd -odometers

Orbit equivalence and speedups of Zd-odometers

Theorem (Johnson-M) Let T and S be Zd−odometers. TFAE:

1 For some cone C ⊆ Zd, T cts

• C S.

2 T and S are continuously orbit equivalent.

Corollary There exists a product-type Zd-odometer T and a non-product-type Zd-odometer S, such that T cts S and S cts T.

David McClendon Speedups of Zd -odometers