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 $f \colon X\rightarrow X$ over a compact metric space $X$ and an asymptotically additive sequence of functions $\Phi = \{\phi_n\colon X\rightarrow \mathbb{R}\}_{n\geq 1}$. If $f$ has the periodic gluing orbit property, then for any constant $a$ satisfying
$\inf\limits_{\mu\in \mathcal M_{inv} (f,X)} \chi_\Phi(\mu) <a<\sup\limits_{\mu\in\mathcal M_{inv} (f,X)} \chi_\Phi(\mu)$
where $\chi_\Phi(\mu) = \liminf_{n\rightarrow \infty}\int\frac1n\phi_n d\mu$, and the infimum and supremum are taken over the set of all $f$-invariant probability measures, there is an ergodic measure $\mu_a\in \mathcal M_{inv} (f,X)$ such that $\chi_\Phi(\mu_a) = a$ and ${\rm{supp}}(\mu)=X.$
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