Markov odometer

In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation is orbit-equivalent to a Markov odometer.^[1]

The basic example of such system is the "nonsingular odometer", based on the additive topological group of p-adic integers endowed with structure of a dynamical system for the transformation $x\mapsto x+{\underline {1}}$ , where ${\underline {1}}:=(1,0,0,\dots )$ . The general form, which is called "Markov odometer", can be constructed through Bratteli–Vershik diagram to define Bratteli–Vershik compactum space together with a corresponding transformation.

Construction

Nonsingular odometer

We introduce first the basic example of dyadic integers. Let $\Omega =\{0,1\}^{\mathbb {N} }$ be the dyadic integers, andowed with the addition which is defined for each coordinate by $(x+y)_{n}=x_{n}+y_{n}+\varepsilon _{n}\,{\bmod {\,}}2$ where $\varepsilon _{0}=0$ and

\varepsilon _{n}={\begin{cases}0&x_{n-1}+y_{n-1}<2\\1&x_{n-1}+y_{n-1}=2\end{cases}}

inductively. Let ${\mathcal {B}}$ be the Borel sigma-algebra generated by the Cylinder set, with the measure $\mu _{p}(x_{n}=1)=p$ and $\mu _{p}(x_{n}=0)=1-p$ , for some $0<p<1$ independently on $n$ . Let $T:\Omega \to \Omega$ be the transformation $T(x)=x+{\underline {1}}$ , where ${\underline {1}}:=(1,0,0,\dots )$ . In other words, $T$ is the transformation $T\left(1,\dots ,1,0,x_{k+1},x_{k+2},\dots \right)=\left(0,\dots ,0,1,x_{k+1},x_{k+2},\dots \right)$ .

One can show that the $\textstyle {\frac {d\mu _{p}\circ T}{d\mu _{p}}}=\left({\frac {1-p}{p}}\right)^{\varphi }$ where $\varphi (x)=\min \left\{n\in \mathbb {N} \mid x_{n}=0\right\}-2$ . Hence $T$ is $\mu _{p}$ -nonsingular. Moreover, one can show that $T$ has the following property: For every $x \in \Omega$ and natural number $n$ , the orbit of $x$ under $T$ along the steps $0,1,\ldots ,2^{n}-1$ is exactly the set $\{0,1\}^{n}$ (as presentation of dyadic integers). As a result, $T$ is $\mu _{p}$ -ergodic. Finally, $T$ is conservative since every invertible ergodic nonsingular transformation in nonatomic space is conservative.

The same construction enables to define such a system for every p-adic integers, and more generally for every product space of the form $\textstyle \Omega =\prod _{n\in \mathbb {N} }A_{n}$ for $A_{n}=\{0,1,\dots ,m_{n}-1\}$ where $m_{n}\in \mathbb {N}$ , endowed with the product Borel sigma-algebra and some nonatomic measure $\textstyle \mu =\prod _{n\in \mathbb {N} }\mu _{n}$ where $\mu _{n}$ is some measure on $A_{n}$ . The corresponding map is defined by $T(x_{1},\dots ,x_{k},x_{k+1},x_{k+2},\dots )=(0,\dots ,0,x_{k+1},x_{k+2},\dots )$ where $k$ is the smallest index for which $x_{k}\neq m_{k}-1$ .

Markov odometer

Let $B=(V,E)$ be an ordered Bratteli–Vershik diagram, consists on a set of vertices of the form $\textstyle \coprod _{n\in \mathbb {N} }V^{(n)}$ (disjoint union) where $V^{0}$ is a singelton and on a set of edges $\textstyle \coprod _{n\in \mathbb {N} }E^{(n)}$ (disjoint union).

The diagram includes source surjection-mappings $s_{n}:E^{(n)}\to V^{(n-1)}$ and range surjection-mappings $r_{n}:E^{(n)}\to V^{(n)}$ . We assume that $e,e'\in E^{(n)}$ are comparable if and only if $r_{n}(e)=r_{n}(e')$ .

For such diagram we look at the product space $\textstyle E:=\prod _{n\in \mathbb {N} }E^{(n)}$ equipped with the product topology. Define "Bratteli–Vershik compactum" to be the subspace of infinite paths,

X_{B}:=\left\{x=(x_{n})_{n\in \mathbb {N} }\in E\mid x_{n}\in E^{(n)}{\text{ and }}r(x_{n})=s(x_{n+1})\right\}

Assume there exists only one infinite path $x_{\max }=(x_{n})_{n\in \mathbb {N} }$ for which each $x_{n}$ is maximal and similarly one infinite path $x_{\text{min}}$ . Define the "Bratteli-Vershik map" $T_{B}:X_{B}\to X_{B}$ by $T(x_{\max })=x_{\min }$ and, for any $x=(x_{n})_{n\in \mathbb {N} }\neq x_{\max }$ define $T_{B}(x_{1},\dots ,x_{k},x_{k+1},\dots )=(y_{1},\dots ,y_{k},x_{k+1},\dots )$ , where $k$ is the first index for which $x_{k}$ is not maximal and accordingly let $(y_{1},\dots ,y_{k})$ be the unique path for which $y_{1},\dots ,y_{k-1}$ are all maximal and $y_{k}$ is the successor of $x_{k}$ . Then $T_{B}$ is homeomorphism of $X_B$ .

Let $P=\left(P^{(1)},P^{(2)},\dots \right)$ be a sequence of stochastic matrices $P^{(n)}=\left(p_{(v,e)\in V^{n-1}\times E^{(}n)}^{(n)}\right)$ such that $p_{v,e}^{(n)}>0$ if and only if $v=s_{n}(e)$ . Define "Markov measure" on the cylinders of $X_B$ by $\mu _{P}([e_{1},\dots ,e_{n}])=p_{s_{1}(e_{1}),e_{1}}^{(1)}\cdots p_{s_{n}(e_{n}),e_{n}}^{(n)}$ . Then the system $\left(X_{B},{\mathcal {B}},\mu _{P},T_{B}\right)$ is called a "Markov odometer".

One can show that the nonsingular odometer is a Markov odometer where all the $V^{(n)}$ are singeltons.

References

↑ A. H. Dooley and T. Hamachi, Nonsingular dynamical systems, Bratteli diagrams and Markov odometers. Isr. J. Math. 138 (2003), 93–123.

Markov odometer

Construction

Nonsingular odometer

Markov odometer

References

Further reading