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Coarse expanding conformal systems were introduced by P. Ha\"issinsky and K.M. Pilgrim to study the essential dynamical properties of certain rational mapson the Riemann sphere in complex dynamics from the point of view of Sullivan'sdictionary.  In this paper, we prove that for a metric coarse expanding conformal system$f\colon(\mathfrak{X}_1,X)\rightarrow (\mathfrak{X}_0,X)$ with repellor $X$,the map $f|_X\colon X\rightarrow X$ is asymptotically $h$-expansive. Moreover,we show that $f|_X$ is not $h$-expansive if there exists at least one branchpoint in the repellor. As a consequence of asymptotic $h$-expansiveness, for$f|_X$ and each real-valued continuous potential on $X$, there exists at leastone equilibrium state. For such maps, if some additional assumptions aresatisfied, we can furthermore establish a level-$2$ large deviation principlefor iterated preimages, followed by an equidistribution result.

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