On the connectedness of limit net sets
The fractal objects considered in this paper are limit net sets described by means of net matrices (see definition below). The limit net sets can also be viewed as the limit set of a Moran construction (subsection 3.3) and from some known results on Moran fractals follows that limit net sets are fractals whose Hausdorff and box-counting dimensions equal \(1+\log 3/\log 4\) . More generally, limit net sets are also a species of random fractals, where the self-similarity property is replaced by statistical self-similarity, that means that enlargements of small parts have the same statistical distribution as the whole set (it can be checked that limit net sets are a particular case of random net fractals introduced by \textit{K. J. Falconer} in [Math. Proc. Camb. Philos. Soc. 100, 559--582 (1986; Zbl 0623.60020)]; see also [\textit{K. J. Falconer}, Fractal Geometry, Mathematical Foundations and Applications. Chichester etc.: John Wiley \& Sons (1990; Zbl 0689.28003)]. First we need to explain some definitions considered in the paper. A matrix \( A_{0}=(a_{i,j}^{0})_{0\leq i,j\leq 3}\) with entries in \(\{0,1\}\) is a net matrix of level \(0\) if exactly \(4\) entries equal \(1\), in each row and in each column of \(A\) there is exactly one entry with value \(1\) and in each submatrix \((a_{i,j})_{i=k,k+1,j=l,l+1}\), where \(k,l\in \{0,2\}\), there is exactly one entry with value \(1\). There are \(16\) net matrices. Two entries of a net matrix are neighbours if either their column indices coincide and their row indices are consecutive integers or their row indices coincide and their column indices are consecutive integers. A path in a net matrix is a finite sequence of neighbouring \(0\) entries of the matrix. A net matrix is said to be connected if for any two \(0\) entries there exists a path in the matrix that connects them. If we substitute in the net matrix \(A_{0}=(a_{i,j}^{0})_{0\leq i,j\leq 3}\) of level \(0\) each entry \(0\) by one net matrix and each entry \(1\) by a \( 4\times 4\) matrix with all its entries equal \(1\) we obtain a net matrix of level \(1\). By induction are defined the net matrices of level \(k\). If we substitute at each stage \(r\) all \(0\) entries by the same net matrix \(A(r)\) then it is said that the net matrices generated has been obtained by a uniform net substitution. If \(A_{k}=(a_{i,j}^{k})_{0\leq i,j\leq 2^{2(k+1)}-1}\) is a net matrix of level \(k\), then \(I_{i,j}^{k}:=I_{i}^{k}\times I_{j}^{k}\), where \(I_{i}^{k}\) is the interval \(\left[ \frac{i}{2^{2(k+1)}},\frac{i+1}{2^{2(k+1)}}\right] \) , is called a net square of level \(k\), and the set \(E_{k}:=\cup \{ I_{i,j}^{k}:a_{i,j}^{k}=0\), \(0\leq i,j\leq 2^{2(k+1)}-1\}\) is called a net set of level \(k\). If this matrix \(A_{k}\) has been obtained by a uniform net substitution then it is said that the corresponding net set \(E_{k}\) is a uniform net set of level \(k\). Two net squares \(I_{i,j}^{k}\) and \(I_{i^{\prime },j^{\prime }}^{k}\) of \(E_{k} \)\ are neighbours if they share and edge (or, equivalently, if the entries \( a_{i,j}^{k}\) and \(a_{i^{\prime },j^{\prime }}^{k}\) are neighbours in the corresponding net matrix \(A_{k}\)). A \(k\)-path in a net set \(E_{k}\) is a finite sequence \(p\)\ of neighbouring net squares of\ \(E_{k}\), whose union is called the corridor of the \(k\)-path \(p\). A net set \(E_{k}\) is strongly connected if for any net squares \(S\) and \(T\) of \(E_{k}\) there exists a \(k\) -path \(p=p(S,T)\) that connects \(S\) and \(T\). This is equivalent to the connectedness of the matrix \(A_{k}\) corresponding to \(E_{k}\). Let \((E_{k})_{k}\) be a decreasing sequence of net sets, \(k\) being the level of \(E_{k}\) for \(k\geqslant 0\). The set \(E_{\infty }:=\cap \{E_{k}:k\geqslant 0\}\) is named the limit net set of the sequence \((E_{k})_{k}\). It is obvious that a limit net set \(E_{\infty }\)\ is a nonvoid compact set. This limit net set \(E_{\infty }\) is said to be net-connected if for each two points \(x\), \( y\in E_{\infty }\) and for all \(k\geqslant 0\) there exists two \(k\)-net squares \(S_{k}\), \(T_{k}\in E_{k}\) with \(x\in S_{k}\) and \(y\in T_{k}\) and a \(k \)-path in \(E_{k}\) that connects \(S_{k}\) and \(T_{k}\). Finally, if the sets \( E_{k}\) of the sequence \((E_{k})_{k}\) are uniform net sets then \(E_{\infty }\) is called a uniform limit net set. Now, after the preceding definitions, we would like to announce some of the results obtained in this paper. It is proved that if \((E_{k})_{k\geqslant 0}\) is a decreasing sequence of strongly connected net sets, where \(k\) is the level of \(E_{k}\) for \(k\geqslant 0\), then the corresponding limit net set \( E_{\infty }\) is net-connected, connected and locally connected with respect to the topology induced by the Euclidean metric on \(\mathbb{R}^{2}\) (Theorems 2, 1 and Proposition 3), therefore \(E_{\infty }\) is a continuous image of an interval, being in particular arcwise connected (Corollary 1). It is proved that the converse of Theorem 2 holds (see Theorem 4, where it is stated that net-connectedness of \(E_{\infty }\) implies that each \(E_{k}\) is strongly connected). Also it is proved that the reciprocal affirmations of the other three results do not hold. To characterize disconnectedness of a uniform limit net set \(E_{\infty }\), the author defines that a net matrix \(A=(a_{i,j})_{0\leq i,j\leq 3}\) is of type I if \((a_{0,3}-1)(a_{3,0}-1)=0\) and that \(A\) is of type II if \( (a_{0,0}-1)(a_{3,3}-1)=0\). In Theorems 5 and 6 of the paper it is stated that if \(E_{\infty }\) is the limit net of a decreasing sequence of uniform net sets \((E_{k})_{k\geqslant 0}\) and if \(A_{0}\), \(A_{1}\),\(\cdots \), \(A_{k}\) denote the sequence of net matrices that defines the set \(E_{k}\) and if \( m_{1}(k)\) (\(m_{2}(k)\)) denote the number of disconnected matrices of type I (of type II, respectively) then \(E_{\infty }\) is totally disconnected with respect to the topology induced by the Euclidean metric on \(\mathbb{R}^{2}\) if and only if \(\lim_{k\rightarrow \infty }m_{i}(k)=\infty \), for \(i=1, \) \(2\). Characterizations of disconnected but not totally disconnected (connected but not net-connected) uniform limit net sets are given in Theorem \(7\) (Theorem \(8\), respectively). The definition and construction method of limit net sets leads in a natural way to the question of net percolation. Remember that the process of fractal percolation was described by Mandelbrot in the 1970s. Its study has been intensified and extended in the 1980s, because fractal percolation has been used as a model for various physical processes such as intermittency in turbulence or distribution of galaxies in the Universe. In this paper the author defines that net percolation in the unit square occurs for a decreasing sequence of net sets \((E_{k})_{k\geqslant 0}\) if there exists a path (i.e., the image of the interval [0,1] by a continuous real function) in the corresponding limit net set \(E_{\infty }\) that connects two opposite sides of the unit square, and she proves that if each \(E_{k}\) is strongly connected then net percolation in the unit square occurs for the sequence \( (E_{k})_{k\geqslant 0}\). With a counterexample the author proves that the above condition is not sufficient for net percolation and then the question of whether and when net percolation occurs in the unit square arises for the sequence \( (E_{k})_{k\geqslant 0}\) in a natural way. In the paper this question is reformulated in terms of random fractals, since the obtained theorems and propositions do not give a complete answer to this question. To this aim it is introduced the limit net set \(E_{\infty }(p)\) of a decreasing sequence \( (E_{k}(p))_{k\geqslant 0}\) of sets obtained with some sequence of net matrices \((A_{k}(p))_{k\geqslant 0}\), where \(A_{k+1}(p)\) is obtained from \( A_{k}(p)\) replacing each zero entry \(a_{ij}\) of \(A_{k}(p)\) by some net matrix \(A(i,j)\), \(p\) being the probability that \(A(i,j)\) is a connected matrix. As in the non probabilistic case, here each \(1\) entry is replaced by a \(4\times 4\) matrix with all its entries equal \(1\). Finally, the following open problem it is proposed in the paper: How does, in terms of probability, the structure of \(E_{\infty }(p)\) vary in dependency of the parameter \(p\)? Does there exist a critical value \(p_{c}\), such that for \(p>p_{c}\) there is positive probability that \(E_{\infty }(p)\) connects opposite sides of the unit square?
scientific article; zbMATH DE number 5346284