On the one-way specification property and large deviations for non-transitive systems

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.