  We introduce a weaker form of the specification property, called "one-way
specification property", and give several examples of non-transitive systems
satisfying this property. As an application, we show that the
$(-\beta)$-transformation satisfies a level-2 large deviation principle with
the Lebesgue measure and the rate function is the free energy under the
condition that $\beta>1$ is a Yrrap number, that is, the orbit of $1$ under the
$(-\beta)$-transformation is eventually periodic.

Dynamical Systems