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On Isomorphism Problem of Stationary Processes
A. H. Zaslavskiĭ Novosibirsk
Abstract:
The central problem in ergodic theory is that of isomorphism. In the paper the sufficient condition for isomorphism of the stationary process
$\xi=(\dots,\xi_{-1},\xi_0,\xi_1,\dots)$,
$\xi_n=0$,
$1,\dots,l$, with some stationary process
$\eta=(\dots,\eta_{-1},\eta_0,\eta_1,\dots)$,
$\eta_n=\alpha_1,\dots,\alpha_m$,
$m\leqq l$, is found. This condition is expressed in terms of a one-dimensional distribution of the process
$\xi$. Isomorphism is constructed with the aid of elementary codes
$$
(i)=\eta_1^i\eta_2^i\cdots\eta_{\omega_i}^i,\qquad i=1,\dots,l,
$$
which code the elementary words
$$
(i)=\underbrace{i00\dots 0}_{\omega_i}.
$$
One of the examples considered proves that it is possible to construct a system of elementary codes for any arbitrary
$l$ and
$m$. This system possesses some properties which secure unique decoding of the sequence
$\eta$.
Received: 31.08.1962