Saturation Reachability Length and the Rate of Convergence of the Local Density of Coherent Structures

The present paper is devoted to the rate of convergence of the local density under transformations of coherent structures in the Medium [4]. We introduce the notions of the saturation reachability length , of a hexagonal cohort, of the Hausdorff metric, and of a hex-converging transformer with rate . Two results are proved. For a hex-converging transformer with rate whose hexagonal cohort is fixed and which does not change the radii, the saturation reachability length is finite for sufficiently large and admits an explicit estimate in terms of , the Hausdorff metric, and the radii of the cells. For a contracting transformer with constant that does not change the radii, under the additional condition that the local density is monotone in the iteration index, the limit of the local density exists and equals . In particular, if , this limit is strictly less than ; in this case, for sufficiently small , the saturation reachability length is infinite if the initial density is below the threshold , and equals zero otherwise. Without the condition , the assertion that the limit does not exceed is false (counterexample: , a one-element cohort).