Intermediate Lyapunov exponents for systems with periodic orbit gluing property

We prove that the average Lyapunov exponents of asymptotically additive functions have the intermediate value property provided the dynamical system has the periodic gluing orbit property. To be precise, consider a continuous map \begin{document}$f \colon X\rightarrow X$\end{document} over a compact metric space \begin{document}$X$\end{document} and an asymptotically additive sequence of functions \begin{document}$\Phi = \{\phi_n\colon X\rightarrow \mathbb{R}\}_{n\geq 1}$\end{document} . If \begin{document}$f$\end{document} has the periodic gluing orbit property, then for any constant \begin{document}$a$\end{document} satisfying \begin{document}$\inf\limits_{\mu\in \mathcal M_{inv} (f,X)} \chi_\Phi(\mu) where \begin{document}$\chi_\Phi(\mu) = \liminf_{n\rightarrow \infty}\int\frac1n\phi_n d\mu$\end{document} , and the infimum and supremum are taken over the set of all \begin{document}$f$\end{document} -invariant probability measures, there is an ergodic measure \begin{document}$\mu_a\in \mathcal M_{inv} (f,X)$\end{document} such that \begin{document}$\chi_\Phi(\mu_a) = a$\end{document} and \begin{document}${\rm{supp}}(\mu)=X.$\end{document}