TY - JOUR

T1 - The dimensions of limits of vertex replacement rules

AU - Previte, Michelle

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2007

Y1 - 2007

N2 - Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R(G). By iterating this procedure on each resulting graph, a sequence of graphs {Rn(G)} is obtained. When the graphs in this sequence are normalized to have diameter one, questions of convergence can be investigated. Sufficient conditions for convergence in the Gromov-Hausdorff metric were given by J. Previte, M. Previte, and M. Vanderschoot for such normalized sequences of graphs when the replacement rule TJ has more than one replacement graph. M. Previte and H.S. Yang showed that under these conditions, the limits of such sequences have topological dimension one. In this paper, we compute the box and Hausdorff dimensions of limit spaces of normalized sequences of iterated vertex replacements when there is more than one replacement graph. Since the limit spaces have topological dimension one and typically have Hausdorff (and box) dimension greater than one, they are fractals. Finally, we give examples of vertex replacement rules that yield fractals.

AB - Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R(G). By iterating this procedure on each resulting graph, a sequence of graphs {Rn(G)} is obtained. When the graphs in this sequence are normalized to have diameter one, questions of convergence can be investigated. Sufficient conditions for convergence in the Gromov-Hausdorff metric were given by J. Previte, M. Previte, and M. Vanderschoot for such normalized sequences of graphs when the replacement rule TJ has more than one replacement graph. M. Previte and H.S. Yang showed that under these conditions, the limits of such sequences have topological dimension one. In this paper, we compute the box and Hausdorff dimensions of limit spaces of normalized sequences of iterated vertex replacements when there is more than one replacement graph. Since the limit spaces have topological dimension one and typically have Hausdorff (and box) dimension greater than one, they are fractals. Finally, we give examples of vertex replacement rules that yield fractals.

UR - http://www.scopus.com/inward/record.url?scp=38749137155&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38749137155&partnerID=8YFLogxK

U2 - 10.1215/ijm/1258138427

DO - 10.1215/ijm/1258138427

M3 - Article

AN - SCOPUS:38749137155

VL - 51

SP - 513

EP - 536

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

SN - 0019-2082

IS - 2

ER -