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# An Example of a Vertex Replacement Rule that gives Convergence as well as Exponential Growth - PowerPoint PPT Presentation

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. Graphs. Definition (roughly): A graph consists of two things Points (vertices)

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that gives Convergence as well as

Exponential Growth

Casey Fye

Penn State Erie, The Behrend College

April 2009

• 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

• 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

v1

v1

H1

w1

w2

w1

w2

v3

v2

w3

G

v2

v3

w3

H2

R(G)

• 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

• 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

G

R(G)

R2(G)

R3(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), …}

Nonreplaceable Marked Point

(G, p0)

(R(G), p1)

(R2(G), p2)

(R3(G), p3)

(R(G), p1)

(G, p0)

(R2(G), p2 )

(R3(G), p3 )

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)

• 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?

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}|

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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

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p

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m-1

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i=1

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• 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

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

• 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?

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

• 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?

• M has polynomial growth with degree equal to the dimension of S

Yes and No

No.

My example:

R

• 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

• 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

• Definition of Box Dimension≈

Definition of Hausdorff Dimension

dimbox(S)=2

dimhaus(S)=∞

The Sequence Rn(G)

R

R(G)

R2(G)

R3(G)