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To put Anthony's example in a more general context, if you are familiar with Vershik maps on Bratteli diagrams and dimension groups, then you can find lots of minimal uniquely ergodic actions on Cantor sets that come from the $Z_2$-orbit space of a minimal action action with two ergodic probability measures, with an involution that interchanges the measures.

More explicitly, begin with a Bratteli diagram representing a simple dimension group with unique trace (for example, the 2-adic odometer is represented by $\lim \times 2: Z\to Z$); to be interesting, lots of the multiplicities should exceed $1$, in fact, they should be quite large (which can be arranged by telescoping). Then we can replace the integer multiplicities by characters (not irreducible of course) of the group $Z_2$, and having degrees equalling the multiplicities. Pick a Vershik adic map, e.g., using the left/right ordering.

This yields a new Bratteli diagram, together with a natural action of $Z_2$, and such that the $Z_2$-orbit space is precisely the original action. Except for degenerate cases, this yields a minimal system with a $Z_2$-action, and necessarily a two to one map. Generically, the new minimal system is uniquely ergodic. However, we can easily construct examples where there are two ergodic measures (corresponding to two pure traces on the dimension group).

For example, if we telescope the 2-adic odometer thingy above, so that at the $n$th level, there are $2^n$ edges to the next point, we can realize a character of $Z_2$ whose dimension is $2^n$, corresponding to the matrix $$ A_n:= \left( \begin{array}{cc} 2^n-1 & 1 \\ 1 & 2^n-1 \end{array} \right). $$ This dimension group, $\lim A_n: Z^2 \to Z_2$ is simple and has two pure traces, and there are lots and lots of similar examples. This particular one can be made to sit over the 2-adic odometer by choosing the ordering for the Vershik map appropriately.

This forms part of the classification of actions of $Z_2$ (and other groups) on AF C*-algebras which leave the algebraic direct limit stable. There are a few papers in this area, but I've gone on long enough.

To put Anthony's example in a more general context, if you are familiar with Vershik maps on Bratteli diagrams and dimension groups, then you can find lots of minimal uniquely ergodic actions on Cantor sets that come from the $Z_2$-orbit space of a minimal action action with two ergodic probability measures, with an involution that interchanges the measures.

More explicitly, begin with a Bratteli diagram representing a simple dimension group with unique trace (for example, the 2-adic odometer is represented by $\lim \times 2: Z\to Z$); to be interesting, lots of the multiplicities should exceed $1$, in fact, they should be quite large (which can be arranged by telescoping). Then we can replace the integer multiplicities by characters (not irreducible of course) of the group $Z_2$, and having degrees equalling the multiplicities. Pick a Vershik adic map, e.g., using the left/right ordering.

This yields a new Bratteli diagram, together with a natural action of $Z_2$, and such that the $Z_2$-orbit space is precisely the original action. Except for degenerate cases, this yields a minimal system with a $Z_2$-action, and necessarily a two to one map. Generically, the new minimal system is uniquely ergodic. However, we can easily construct examples where there are two ergodic measures (corresponding to two pure traces on the dimension group).

For example, if we telescope the 2-adic odometer thingy above, so that at the $n$th level, there are $2^n$ edges to the next point, we can realize a character of $Z_2$ whose dimension is $2^n$, corresponding to the matrix $$ A_n:= \left( \begin{array}{cc} 2^n-1 & 1 \\ 1 & 2^n-1 \end{array} \right). $$ This dimension group, $\lim A_n: Z^2 \to Z^2$ is simple and has two pure traces, and there are lots and lots of similar examples. This particular one can be made to sit over the 2-adic odometer by choosing the ordering for the Vershik map appropriately.

This forms part of the classification of actions of $Z_2$ (and other groups) on AF C*-algebras which leave the algebraic direct limit stable. There are a few papers in this area, but I've gone on long enough.

To put Anthony's example in a more general context, if you are familiar with Vershik maps on Bratteli diagrams and dimension groups, then you can find lots of minimal uniquely ergodic actions on Cantor sets that come from the $Z_2$-orbit space of a minimal action action with two ergodic probability measures, with an involution that interchanges the measures.

More explicitly, begin with a Bratteli diagram representing a simple dimension group with unique trace (for example, the 2-adic odometer is represented by $\lim \times 2: Z\to Z$); to be interesting, lots of the multiplicities should exceed $1$, in fact, they should be quite large (which can be arranged by telescoping). Then we can replace the integer multiplicities by characters (not irreducible of course) of the group $Z_2$. Pick a Vershik adic map, e.g., using the left/right ordering.

This yields a new Bratteli diagram, together with a natural action of $Z_2$, and such that the $Z_2$-orbit space is precisely the original action. Except for degenerate cases, this yields a minimal system with a $Z_2$-action, and necessarily a two to one map. Generically, the new minimal system is uniquely ergodic. However, we can easily construct examples where there are two ergodic measures (corresponding to two pure traces on the dimension group).

For example, if we telescope the 2-adic odometer thingy above, so that at the $n$th level, there are $2^n$ edges to the next point, we can realize a character of $Z_2$ whose dimension is $2^n$, corresponding to the matrix $$ A_n:= \left( \begin{array}{cc} 2^n-1 & 1 \\ 1 & 2^n-1 \end{array} \right). $$ This dimension group, $\lim A_n: Z^2 \to Z_2$ is simple and has two pure traces, and there are lots and lots of similar examples. This particular one can be made to sit over the 2-adic odometer by choosing the ordering for the Vershik map appropriately.

This forms part of the classification of actions of $Z_2$ (and other groups) on AF C*-algebras which leave the algebraic direct limit stable. There are a few papers in this area, but I've gone on long enough.

To put Anthony's example in a more general context, if you are familiar with Vershik maps on Bratteli diagrams and dimension groups, then you can find lots of minimal uniquely ergodic actions on Cantor sets that come from the $Z_2$-orbit space of a minimal action action with two ergodic probability measures, with an involution that interchanges the measures.

More explicitly, begin with a Bratteli diagram representing a simple dimension group with unique trace (for example, the 2-adic odometer is represented by $\lim \times 2: Z\to Z$); to be interesting, lots of the multiplicities should exceed $1$, in fact, they should be quite large (which can be arranged by telescoping). Then we can replace the integer multiplicities by characters (not irreducible of course) of the group $Z_2$, and having degrees equalling the multiplicities. Pick a Vershik adic map, e.g., using the left/right ordering.

This yields a new Bratteli diagram, together with a natural action of $Z_2$, and such that the $Z_2$-orbit space is precisely the original action. Except for degenerate cases, this yields a minimal system with a $Z_2$-action, and necessarily a two to one map. Generically, the new minimal system is uniquely ergodic. However, we can easily construct examples where there are two ergodic measures (corresponding to two pure traces on the dimension group).

For example, if we telescope the 2-adic odometer thingy above, so that at the $n$th level, there are $2^n$ edges to the next point, we can realize a character of $Z_2$ whose dimension is $2^n$, corresponding to the matrix $$ A_n:= \left( \begin{array}{cc} 2^n-1 & 1 \\ 1 & 2^n-1 \end{array} \right). $$ This dimension group, $\lim A_n: Z^2 \to Z_2$ is simple and has two pure traces, and there are lots and lots of similar examples. This particular one can be made to sit over the 2-adic odometer by choosing the ordering for the Vershik map appropriately.

This forms part of the classification of actions of $Z_2$ (and other groups) on AF C*-algebras which leave the algebraic direct limit stable. There are a few papers in this area, but I've gone on long enough.

To put Anthony's example in a more general context, if you are familiar with Vershik maps on Bratteli diagrams and dimension groups, then you can find lots of minimal uniquely ergodic actions on Cantor sets that come from the $Z_2$-orbit space of a minimal action action with two ergodic probability measures, with an involution that interchanges the measures.

More explicitly, begin with a Bratteli diagram representing a simple dimension group with unique trace (for example, the 2-adic odometer is represented by $\lim \times 2: Z\to Z$); to be interesting, lots of the multiplicities should exceed $1$, in fact, they should be quite large (which can be arranged by telescoping). Then we can replace the integer multiplicities by characters (not irreducible of course) of the group $Z_2$. Pick a Vershik adic map, e.g., using the left/right ordering.

This yields a new Bratteli diagram, together with a natural action of $Z_2$, and such that the $Z_2$-orbit space is precisely the original action. Except for degenerate cases, this yields a minimal system with a $Z_2$-action, and necessarily a two to one map. Generically, the new minimal system is uniquely ergodic. However, we can easily construct examples where there are two ergodic measures (corresponding to two pure traces on the dimension group).

For example, if we telescope the 2-adic odometer thingy above, so that at the $n$th level, there are $2^n$ edges to the next point, we can realize a character of $Z_2$ whose dimension is $2^n$, corresponding to the matrix $$ A_n:= \left( \begin{array}{cc} 2^n-1 & 1 \\ 2^n-1 & 1 \end{array} \right). $$ This dimension group, $\lim A_n: Z^2 \to Z_2$ is simple and has two pure traces, and there are lots and lots of similar examples. This particular one can be made to sit over the 2-adic odometer by choosing the ordering for the Vershik map appropriately.

This forms part of the classification of actions of $Z_2$ (and other groups) on AF C*-algebras which leave the algebraic direct limit stable. There are a few papers in this area, but I've gone on long enough.

To put Anthony's example in a more general context, if you are familiar with Vershik maps on Bratteli diagrams and dimension groups, then you can find lots of minimal uniquely ergodic actions on Cantor sets that come from the $Z_2$-orbit space of a minimal action action with two ergodic probability measures, with an involution that interchanges the measures.

More explicitly, begin with a Bratteli diagram representing a simple dimension group with unique trace (for example, the 2-adic odometer is represented by $\lim \times 2: Z\to Z$); to be interesting, lots of the multiplicities should exceed $1$, in fact, they should be quite large (which can be arranged by telescoping). Then we can replace the integer multiplicities by characters (not irreducible of course) of the group $Z_2$. Pick a Vershik adic map, e.g., using the left/right ordering.

This yields a new Bratteli diagram, together with a natural action of $Z_2$, and such that the $Z_2$-orbit space is precisely the original action. Except for degenerate cases, this yields a minimal system with a $Z_2$-action, and necessarily a two to one map. Generically, the new minimal system is uniquely ergodic. However, we can easily construct examples where there are two ergodic measures (corresponding to two pure traces on the dimension group).

For example, if we telescope the 2-adic odometer thingy above, so that at the $n$th level, there are $2^n$ edges to the next point, we can realize a character of $Z_2$ whose dimension is $2^n$, corresponding to the matrix $$ A_n:= \left( \begin{array}{cc} 2^n-1 & 1 \\ 1 & 2^n-1 \end{array} \right). $$ This dimension group, $\lim A_n: Z^2 \to Z_2$ is simple and has two pure traces, and there are lots and lots of similar examples. This particular one can be made to sit over the 2-adic odometer by choosing the ordering for the Vershik map appropriately.

This forms part of the classification of actions of $Z_2$ (and other groups) on AF C*-algebras which leave the algebraic direct limit stable. There are a few papers in this area, but I've gone on long enough.