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

Coarse expanding conformal systems were introduced by P. Ha\"issinsky and K. M. Pilgrim to study the essential dynamical properties of certain rational maps on the Riemann sphere in complex dynamics from the point of view of Sullivan's dictionary. 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 branch point in the repellor. We also prove that $f|_X$ is forward expansive when there is no branch point in the repellor. Our study of expansion properties does not rely on symbolic codings or Markov partitions, and exploits directly the geometric data. 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.