We define a quasiconformal metric on the space of infinite signals \({X}^{\mathbb{N}}\) , where X is a finite alphabet, such that the left shift mapping on \({X}^{\mathbb{N}}\) locally looks like a similarity. For each signal the shift iterations generate a sequence of empirical measures on \({X}^{\mathbb{N}}\) and the corresponding set of cluster points. The whole \({X}^{\mathbb{N}}\) is partitioned into narrow basins consisting of signals having the same cluster sets for the sequences of empirical measures, and for each narrow basin we calculate its packing dimension with respect to the quasiconformal metric.
