  Given an automaton (a letter-to-letter transducer, a dynamical 1-Lipschitz
system on the space $\mathbb Z_p$ of $p$-adic integers) $\mathfrak A$ whose
input and output alphabets are $\mathbb F_p=\{0,1,\ldots,p-1\}$, one visualizes
word transformations performed by $\mathfrak A$ by a point set $\mathbf
P(\mathfrak A)$ in real plane $\mathbb R^2$.
  For a finite-state automaton $\mathfrak A$, it is shown that once some points
of $\mathbf P(\mathfrak A)$ constitute a smooth (of a class $C^2$) curve in
$\mathbb R^2$, the curve is a segment of a straight line with a rational slope;
and there are only finitely many straight lines whose segments are in
$\mathbf{P}(\mathfrak A)$. Moreover, when identifying $\mathbf P(\mathfrak A)$
with a subset of a 2-dimensional torus $\mathbb T^2\subset\mathbb R^3$ (under a
natural mapping of the real unit square $[0,1]^2$ onto $\mathbb T^2$) the
smooth curves from $\mathbf P(\mathfrak A)$ constitute a collection of torus
windings. In cylindrical coordinates either of the windings can be ascribed to
a complex-valued function $\psi(x)=e^{i(Ax-2\pi B(t))}$ $(x\in\mathbb R)$ for
suitable rational $A,B(t)$. Since $\psi(x)$ is a standard expression for a
matter wave in quantum theory (where $B(t)=tB(t_0)$), and since transducers can
be regarded as a mathematical formalization for causal discrete systems, the
paper might serve as a mathematical reasoning why wave phenomena are inherent
in quantum systems: This is because of causality principle and the discreteness
of matter.
