  To characterize local finite-time properties associated with transient chaos
in open dynamical systems, we introduce an escape rate and fractal dimensions
suitable for this purpose in a coarse-grained description. We numerically
illustrate that these quantifiers have a considerable spread across the domain
of the dynamics, but their spatial variation, especially on long but
non-asymptotic integration times, is approximately consistent with the
relationship that was recognized by Kantz and Grassberger for temporally
asymptotic quantifiers. In particular, deviations from this relationship are
smaller than differences between various locations, which confirms the
existence of such a dynamical law and the suitability of our quantifiers to
represent underlying dynamical properties in the non-asymptotic regime.
