  We consider Cantor real numeration system as a frame in which every
non-negative real number has a positional representation. The system is defined
using a bi-infinite sequence $\Beta=(\beta_n)_{n\in\Z}$ of real numbers greater
than one. We introduce the set of $\Beta$-integers and code the sequence of
gaps between consecutive $\Beta$-integers by a symbolic sequence in general
over the alphabet $\N$. We show that this sequence is $S$-adic. We focus on
alternate base systems, where the sequence $\Beta$ of bases is periodic and
characterize alternate bases $\Beta$, in which $\Beta$-integers can be coded
using a symbolic sequence $v_{\Beta}$ over a finite alphabet. With these
so-called Parry alternate bases we associate some substitutions and show that
$v_\Beta$ is a fixed point of their composition. The paper generalizes results
of Fabre and Burd\'ik et al.\ obtained for the R\'enyi numerations systems,
i.e., in the case when the Cantor base $\Beta$ is a constant sequence.
