“The Sierpinski gasket minus its bottom line” as a tree of Sierpinski gaskets. (English) Zbl 1542.31004
This paper focuses primarily on a subset of the Sierpinski gasket, if the three corners of the Sierpinski gasket \(K\) are \((0,0)\), \((1,0)\), and \(\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)\) the authors consider the subset of the gasket where the line segment \(I\) from \((0,0)\) to \((1,0)\) is removed. The main observation is that \(K \setminus I\) is a “tree of Sierpinski gaskets”.
Firstly, the intrinsic metric geometry of \(K \setminus I\) is identical to \(K\) except when close to \(I\). It is shown first that the completion of \(K \setminus I\) with the shortest path metric is homeomorphic to \(K \setminus I \cup \Sigma_T = K\) where \(\Sigma_T\) is a Cantor set and the metric is the usual metric on \(K\) and a \(2\)-adic metric on CS (Theorem 2.15. This is proven by decomposing \(K \setminus I\) into a tree \(T\) of standard Sierpinski gaskets \(K_{\omega}\) and considering when the shortest path between two points would have to venture close to \(I\) or stays away from \(I\).
The next step is to consider the existence of a resistance form on \(K \setminus I\). Again because of the “tree of gaskets” geometry a resistance form can be constructed and domain identified on \(K \setminus I\) as restriction of the standard resistance form on \(K\). Under this construction a resistance form does indeed exist and it generates a resistance metric on \(K \setminus I\). The completion of \(K \setminus I\) under the resistance metric is homeomorphic to the completion of \(K \setminus I\) under the shortest path metric (Theorem 4.5).
The last main result of the paper is an equality between two resistance forms on \(K \setminus I\). The first resistance form is the restriction of the standard resistance \(\mathcal{E}\) form on a Sierpinski gasket. The second resistance form \(\tilde{\mathcal{E}}\) is a resistance form that is approximated by considering functions on graphs approximating \(K \setminus I\) (the same as the graph approximations to \(K\) but with the bottom line removed). Theorem 5.4 provides that when \(f\) on \(K\) in the domain of \(\mathcal{E}\) is “lifted” by \(\rho: K \setminus I \cup CS \rightarrow K\) to a function on \(K \setminus I \cup CS\) then \(\mathcal{E}(f,f) = \tilde{\mathcal{E}}(f \circ \rho, f \circ \rho)\). Thus it can be said that from the resistance perspective \(K\) and \(K \setminus I\) are very closely related.
Firstly, the intrinsic metric geometry of \(K \setminus I\) is identical to \(K\) except when close to \(I\). It is shown first that the completion of \(K \setminus I\) with the shortest path metric is homeomorphic to \(K \setminus I \cup \Sigma_T = K\) where \(\Sigma_T\) is a Cantor set and the metric is the usual metric on \(K\) and a \(2\)-adic metric on CS (Theorem 2.15. This is proven by decomposing \(K \setminus I\) into a tree \(T\) of standard Sierpinski gaskets \(K_{\omega}\) and considering when the shortest path between two points would have to venture close to \(I\) or stays away from \(I\).
The next step is to consider the existence of a resistance form on \(K \setminus I\). Again because of the “tree of gaskets” geometry a resistance form can be constructed and domain identified on \(K \setminus I\) as restriction of the standard resistance form on \(K\). Under this construction a resistance form does indeed exist and it generates a resistance metric on \(K \setminus I\). The completion of \(K \setminus I\) under the resistance metric is homeomorphic to the completion of \(K \setminus I\) under the shortest path metric (Theorem 4.5).
The last main result of the paper is an equality between two resistance forms on \(K \setminus I\). The first resistance form is the restriction of the standard resistance \(\mathcal{E}\) form on a Sierpinski gasket. The second resistance form \(\tilde{\mathcal{E}}\) is a resistance form that is approximated by considering functions on graphs approximating \(K \setminus I\) (the same as the graph approximations to \(K\) but with the bottom line removed). Theorem 5.4 provides that when \(f\) on \(K\) in the domain of \(\mathcal{E}\) is “lifted” by \(\rho: K \setminus I \cup CS \rightarrow K\) to a function on \(K \setminus I \cup CS\) then \(\mathcal{E}(f,f) = \tilde{\mathcal{E}}(f \circ \rho, f \circ \rho)\). Thus it can be said that from the resistance perspective \(K\) and \(K \setminus I\) are very closely related.
Reviewer: Benjamin Steinhurst (Westminster)
MSC:
| 31E05 | Potential theory on fractals and metric spaces |
| 31C25 | Dirichlet forms |
| 28A80 | Fractals |
| 60J45 | Probabilistic potential theory |
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