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edited May 27, 2010 at 19:26

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There is a book by Denker, Grillenberger and Sigmund which deals extensively with this topic: they prove a whole range of theorems which construct subshifts whose invariant measures have specified properties. In particular they construct a mixing subshift with multiple invariant measures of maximal entropy, but this is just one of many interesting constructions.

Subshifts which are not of finite type are, as a general class, quite well understood. In terms of their measurable ergodic theory, they can do pretty much anything which is not prevented by entropy. An example is the following theorem of Jewett. Suppose that $T$ is an ergodic measure-preserving transformation of a probability space $(Z,\mathcal{F},\mu)$, and the entropy of $T$ with respect to $\mu$ is finite. Let $p$ be an integer such that $\log p$ is strictly greater than the entropy of $T$ with respect to $\mu$. Then there is a subshift $X$ of $\Sigma_p$, having a unique invariant measure $\nu$, such that the measure-preserving transformation $(X,\mathcal{B},\nu,\sigma)$ is measurably isomorphic to the transformation $(Z,\mathcal{F},\mu,T)$. (Here $\mathcal{B}$ of course denotes the Borel sigma-algebra of $X$.)

There is a book by Denker, Grillenberger and Sigmund which deals extensively with this topic: they prove a whole range of theorems which construct subshifts whose invariant measures have specified properties. In particular they construct a subshift with multiple invariant measures of maximal entropy, but this is just one of many interesting constructions.

Subshifts which are not of finite type are, as a general class, quite well understood. In terms of their measurable ergodic theory, they can do pretty much anything which is not prevented by entropy. An example is the following theorem of Jewett. Suppose that $T$ is an ergodic measure-preserving transformation of a probability space $(Z,\mathcal{F},\mu)$, and the entropy of $T$ with respect to $\mu$ is finite. Let $p$ be an integer such that $\log p$ is strictly greater than the entropy of $T$ with respect to $\mu$. Then there is a subshift $X$ of $\Sigma_p$, having a unique invariant measure $\nu$, such that the measure-preserving transformation $(X,\mathcal{B},\nu,\sigma)$ is measurably isomorphic to the transformation $(Z,\mathcal{F},\mu,T)$. (Here $\mathcal{B}$ of course denotes the Borel sigma-algebra of $X$.)

There is a book by Denker, Grillenberger and Sigmund which deals extensively with this topic: they prove a whole range of theorems which construct subshifts whose invariant measures have specified properties. In particular they construct a mixing subshift with multiple invariant measures of maximal entropy, but this is just one of many interesting constructions.

Subshifts which are not of finite type are, as a general class, quite well understood. In terms of their measurable ergodic theory, they can do pretty much anything which is not prevented by entropy. An example is the following theorem of Jewett. Suppose that $T$ is an ergodic measure-preserving transformation of a probability space $(Z,\mathcal{F},\mu)$, and the entropy of $T$ with respect to $\mu$ is finite. Let $p$ be an integer such that $\log p$ is strictly greater than the entropy of $T$ with respect to $\mu$. Then there is a subshift $X$ of $\Sigma_p$, having a unique invariant measure $\nu$, such that the measure-preserving transformation $(X,\mathcal{B},\nu,\sigma)$ is measurably isomorphic to the transformation $(Z,\mathcal{F},\mu,T)$. (Here $\mathcal{B}$ of course denotes the Borel sigma-algebra of $X$.)

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created May 27, 2010 at 19:18

6.2k
2
31
64

There is a book by Denker, Grillenberger and Sigmund which deals extensively with this topic: they prove a whole range of theorems which construct subshifts whose invariant measures have specified properties. In particular they construct a subshift with multiple invariant measures of maximal entropy, but this is just one of many interesting constructions.

Subshifts which are not of finite type are, as a general class, quite well understood. In terms of their measurable ergodic theory, they can do pretty much anything which is not prevented by entropy. An example is the following theorem of Jewett. Suppose that $T$ is an ergodic measure-preserving transformation of a probability space $(Z,\mathcal{F},\mu)$, and the entropy of $T$ with respect to $\mu$ is finite. Let $p$ be an integer such that $\log p$ is strictly greater than the entropy of $T$ with respect to $\mu$. Then there is a subshift $X$ of $\Sigma_p$, having a unique invariant measure $\nu$, such that the measure-preserving transformation $(X,\mathcal{B},\nu,\sigma)$ is measurably isomorphic to the transformation $(Z,\mathcal{F},\mu,T)$. (Here $\mathcal{B}$ of course denotes the Borel sigma-algebra of $X$.)