An Example of a Vertex Replacement Rule that gives Convergence as well as Exponential Growth

1. An Example of a Vertex Replacement Rule that gives Convergence as well as Exponential Growth Casey Fye Advisor: Dr. Michelle Previte Penn State Erie, The Behrend College April 2009

2. Graphs • Definition (roughly): A graph consists of two things • Points (vertices) • line segments (called edges) • the endpoints are in the set of vertices • We will consider only graphs whose edges have the same length vertex edge

3. Vertex Replacement Rules • A vertex replacement rule Ris a finite set of finite graphs called replacement graphs, satisfying two conditions. • Each replacement graph has a designated set of vertices called boundary vertices H1 H2

4. A Replacement Rule Acts on G v1 v1 H1 w1 w2 w1 w2 v3 v2 w3 G v2 v3 w3 H2 R(G)

5. Symmetric Condition • A vertex replacement rule R is a finite set of finite graphs called replacement graphs • Each Hi must be symmetric with respect to its boundary vertices

6. Distinct Number of Boundary Vertices Condition • A vertex replacement rule R is a finite set of finite graphs called replacement graphs • Each Hi must be symmetric with respect to its boundary vertices • Each replacement graph Hi has a distinct number of boundary vertices H1 H2 G

7. R(G) has Replaceable Vertices R(G)

8. Sequence of Replacement Graphs G R(G) R2(G) R3(G)

9. Two Possible Options for Studying {Rn(G)} • Allow Rn(G) to grow as n→∞ and designate a marked point, pn for center of reference in Rn(G) • {(G, p0), (R(G), p1), (R2(G), p2), (R3(G), p3), …}

10. Nonreplaceable Marked Point (G, p0) (R(G), p1) (R2(G), p2) (R3(G), p3)

11. Limit of Sequence of Marked Graphs

12. Another Example of a Sequence of Marked Graphs (R(G), p1) (G, p0) (R2(G), p2 ) (R3(G), p3 )

13. Limit of this Sequence of Marked Graphs

14. Two Possible Options for Studying {Rn(G)} 2. Scale each graph in the sequence to the same size as the initial graph • {(G,1), (R(G),1), (R2(G),1), (R3(G),1), …} Limit of (Rn(G), 1) (G, 1) (R(G), 1) (R2(G), 1)

15. What is the Relationship? • Limit M of (Rn(G), pn) What is its growth? • Limit S of (Rn(G), 1) What is its dimension? What is the relationship between growth of M and dimension of S?

16. Growth of the Limit of (Rn(G), pn) • The growth function of a locally finite graph G with respect to a vertex p єV(G) is given by f(G, p, m) = |{y єV(G): d(p,y) ≤ m, 0 ≤ m}|

17. Binary Tree . . . . . . . . . . . . . . . . . Growth Function: f(G, p, 0)= 20 f(G, p, 1)= 20+20+21 f(G, p, 2)= 20+20+21+21+22 f(G, p, m)= 2m+1+∑2i =2m(3)-2 . . . . . p . . . . . . m-1 . . . . i=1 . . . . . . . .

18. Exponential Growth of the Limit of (Rn(G), pn) • The growth function of a locally finite graph G with respect to a vertex p єV(G) is given by f(G, p, m) = |{y єV(G): d(p,y) ≤ m, 0 ≤ m}| • We say G has exponential growth if f(G, p, m)≥cam for some constants c>0 and a>1 • We say G has polynomial growth if f(G, p, m) is bounded above by a polynomial

19. Theorem for Exponential Growth(J. Previte and M. Previte) • Let (G, p0) be an initial marked graph with at least one replaceable vertex, and let R={H} be a replacement rule such that H has a replaceable vertex but the boundary vertices of H are not replaceable. Suppose that there exist two replaceable vertices v1 and v2 in H and nonreplaceable path in H between v1 and v2 that contains exactly one boundary vertex. Then each graph in the set of limit graphs R∞(G, p∞) of (G, p0) has exponential growth. v1 v2

20. What is the Relationship? • Limit M of (Rn(G), pn) What is its growth? • Limit S of (Rn(G), 1) What is its dimension? What is the relationship between growth of M and dimension of S?

21. Theorem for Convergence(J. Previte) • Let H define a vertex replacement rule R and let G be a finite graph with at least one replaceable vertex. Then the normalized sequence {(Rn(G), 1)} converges in the Gromov-Hausdorff metric if and only if one of the following hold: • H contains exactly one replaceable vertex • |∂H|> 1 and every path in H connecting two different boundary vertices contains at least two replaceable vertices Convergence Convergence Divergence H H

22. What is the Relationship? • Limit M of (Rn(G), pn) What is its growth? • Limit S of (Rn(G), 1) What is its dimension? • Conjecture: M has polynomial growth with degree equal to the dimension of S What is the relationship between growth of M and dimension of S?

23. Conjecture • M has polynomial growth with degree equal to the dimension of S Yes and No No. My example: R

24. Exponential Growth • Let (G, p0) be an initial marked graph with at least one replaceable vertex, and let R={H} be a replacement rule such that H has a replaceable vertex but the boundary vertices of H are not replaceable. Suppose that there exist two replaceable vertices v1 and v2 in H and nonreplaceable path in H between v1 and v2 that contains exactly one boundary vertex. Then each graph in the set of limit graphs R∞(G, p∞) of (G, p0) has exponential growth. v1 v2

25. Conjecture • M has polynomial growth with degree equal to the dimension of S Yes Dr.’s Joe and Michelle Previte proved growth of M= dimension of S

26. Important Application of My Example • Definition of Box Dimension≈ Definition of Hausdorff Dimension dimbox(S)=2 dimhaus(S)=∞

27. The Sequence Rn(G) R R(G) R2(G) R3(G)

28. Thank you!