Speedups of Z d -odometers David M. McClendon Ferris State University Big Rapids, MI, USA joint with Aimee S.A. Johnson (Swarthmore) Speedups of Z d -odometers David McClendon
Some words about Jane Speedups of Z d -odometers David McClendon
This talk is about actions of Z d Definition A Z d − measure-preserving system ( Z d -m.p.s. ) is a quadruple ( X , X , µ, T ) where ( X , X , µ ) is a Lebsegue probability space and T = { T v : v ∈ Z d } is an action of Z d on X by measure-preserving transformations. Definition A Z d − Cantor minimal system ( Z d -C.m.s. ) is a pair ( X , T ) where X is a Cantor space and T = { T v : v ∈ Z d } is a minimal action of Z d on X by homeomorphisms. In either situation, we can write T = ( T 1 , ..., T d ) where T j is shorthand for the action of standard basis vector e j . Speedups of Z d -odometers David McClendon
Speedups of actions of Z d Definition A cone C is the intersection of Z d − { 0 } with any open, connected subset of R 2 bounded by d distinct hyperplanes passing through the origin. ✡ • • • • • • • ✡ Example in Z 2 : C • • • • ✡ • • • ✡ • • • • • • • ✡ • • ✡ • • • • • ✏✏✏✏✏✏✏✏✏✏ ✡ • • • • • • • • • • • ✡ • ✡ • • • • • • • • ✡ • • • • • • • Speedups of Z d -odometers David McClendon
Speedups of actions of Z d Definition A C − speedup of Z d − action T = ( T 1 , ..., T d ) is another Z 2 − action T p = ( T 1 , ... T d ) (defined on the same space as T ) such that T j ( x ) = T p j ( x ) ( x ) for some function p = ( p 1 , ..., p d ) : X → C d . 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 T j independently to obtain a speedup of T ). Speedups of Z d -odometers David McClendon
A picture to explain ( d = 2) ✘ ✘✘✘✘ ✘✘ ✘ ✘ ✘ ✘✘✘✘✘✘✘✘✘ ✘✘✘✘✘✘✘ ✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘ ✘✘✘✘✘ ✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ C ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✄ ✗ ✄ ✘ ✲ ✲ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✲ ✲ ✲ ✲ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✿ • • • ✘✘✘✘✘✘✘✘✘✘✘ • • • • • • ✄ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✒ � � ✒ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✻ ✻ ✻ T 2 ✻ T 1 ✻ ✻ ✻ ✘ ✄ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ � � ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✄ � ✲ ✲ ✲ ✲ ✲ ✲ � ✘ T 2 ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ • • • • • • • • • ✘ ✟ ✯ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✄ ✟✟✟✟✟ � ✘ ✁ ✕ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✄ ✁ � ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ T 2 T 1 ✿ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ ✲ ✄ ✲ ✁ ✲ ✲ ✲ ✲ � ✶ ✏ • • • • • • • • ✏✏✏✏✏✏✏✏ ✄ ✁ T 1 ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✄ ✁ ✲ ✁ ✲ ✲ ✲ ✲ ✲ ✄ • • • • • • • • x ✻ ✻ ✻ ✻ ✻ ✻ ✻ T 2 ✲ ✲ ✲ ✲ ✲ ✲ • • • • • • • T 1 Here, T p = ( T 1 , T 2 ) is a C − speedup of T = ( T 1 , T 2 ). In particular, for the indicated point x , we have p ( x ) = ((3 , 1) , (1 , 2)) . Speedups of Z d -odometers David McClendon
� � � � � � � � � 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 p T p T p T T T T T T T T � · · · � • � • · · · • � • • • • T p T p Speedups of Z d -odometers David McClendon
The big picture Question Given two Z d -actions ( X , T ) and ( Y , S ), when is there a speedup of T isomorphic to S ? The word isomorphic means: measurably conjugate, if T and S are Z d -m.p.s. topologically conjugate, if T and S are Z d -C.m.s. Notation Write T � C S if there is a C -speedup of T isomorphic to S . Speedups of Z d -odometers David McClendon
The big picture Question Given two Z d -actions ( X , T ) and ( Y , S ), when is there a speedup of T isomorphic to S ? The word isomorphic means: measurably conjugate, if T and S are Z d -m.p.s. topologically conjugate, if T and S are Z d -C.m.s. Notation Write T � S if for any cone C ⊆ Z d , T � C S . Speedups of Z d -odometers David McClendon
The big picture Question Given two Z d -actions ( X , T ) and ( Y , S ), when is there a speedup of T isomorphic to S ? The word isomorphic means: measurably conjugate, if T and S are Z d -m.p.s. topologically conjugate, if T and S are Z d -C.m.s. Notation Write adjective T S � if T � S via a speedup function p which is that adjective. Speedups of Z d -odometers David McClendon
History of speedups: ergodic theory Theorem (Neveu 1969) integrable Suppose ( X , T ) and ( Y , S ) are m.p.t.s. If T S , then � �� � h ( S ) = p d µ 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 Z d -m.p.s., where T is ergodic. Then T mble � S . Speedups of Z d -odometers David McClendon
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 Z d -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. Speedups of Z d -odometers David McClendon
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. Speedups of Z d -odometers David McClendon
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 Z d -odometers? Speedups of Z d -odometers David McClendon
Z d -odometers Z d -odometers were introduced by Cortez in 2004. They are defined as follows: The phase space Let Z d ≥ G 0 ≥ G 1 ≥ G 2 ≥ G 3 ≥ · · · be a decreasing sequence of subgroups of Z d , each of which have ∞ finite index in Z d , such that j =0 G j = { 0 } . Let X be the inverse ∩ limit − ( Z d / G j ) . X = lim ← Speedups of Z d -odometers David McClendon
Z d -odometers Z d -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 = ( x 0 + G 0 , x 1 + G 1 , x 2 + G 2 , ... ) where the x j are “commensurate”, i.e. since G j ≥ G j +1 , there’s a natural map π j : Z d / G j +1 → Z d / G j ; for such a sequence to be in X we require that, for all j , π j ( x j +1 + G j +1 ) = x j + G j . Speedups of Z d -odometers David McClendon
Z d -odometers Z d -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 j th coordinate is the usual (vector) addition in the quotient group Z d / G j . Given any v ∈ Z d , we can “convert” v into an element of X by setting τ ( v ) = ( v + G 0 , v + G 1 , v + G 2 , ... ) Define the action T of Z d on X by T v ( x ) = x + τ ( v ). ( X , T ) is a Z d -C.m.s. called a Z d -odometer . Speedups of Z d -odometers David McClendon
Speedups of Z d − odometers Theorem (Johnson-M) Let T be a Z d − odometer, and suppose T cts � S . If S is minimal, then S is topologically conjugate to a Z d − odometer. Same result as d = 1 (AAO), but not a similar proof. Speedups of Z d -odometers David McClendon
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