Weak expansion properties and a large deviation principle for coarse expanding conformal systems

In this paper, we prove that for a metric coarse expanding conformal system $f\:(\mathfrak{X}_1,X)\rightarrow (\mathfrak{X}_0,X)$ with repellor $X$, the map $f|_X\:X\rightarrow X$ is asymptotically $h$-expansive. Moreover, we show that $f|_X$ is not $h$-expansive if there exists at least one branch point in the repellor. As a consequence of asymptotic $h$-expansiveness, for $f|_X$ and each real-valued continuous potential on $X$, there exists at least one equilibrium state. For such maps, if some additional assumptions are satisfied, we can furthermore establish a level-2 large deviation principle for iterated preimages, followed by an equidistribution result.