Calculated The Hausdorff Dimension Of Connected Component Of Two Kinds Of Fractal Squares Based On GIFS

This paper mainly discussed two kinds of fractal squares and their Hausdorff dimensions of connected components respectively.These two kinds of fractal squares are any three sides of [0,1]~2 square and any adjacent two sides of[0,1]~2 square.Firstly,E is determined by n and the digital set D,of which E is a fractal square.Secondly,for a fractal square with any three sides of [0,1]~2 square,a GIFS is constructed by defining t.wo label maps h_X and h_Y.It is proved that the connected component C of E with(0,0)is equal to one of the invariant sets of the GIFS.Moreover,according to the Perron-Frobenius theorem,the dimensions of C is equal to log?/logn where ? is the maximum eigenvalue of the corresponding matrix.Thirdly,for a fractal square with any adjacent two sides of[0,1]~2 square,there are also two cases.One is four vertices of[0,1]~2 square connected,the other is disconnected.For four disconnected vertices,the definition of H-connected component is given,A GIFS is constructed by defining four label maps h_A,h_B,h_C and h_D.Furthermore,it also can be proved that the H-connected component F of E with(0,0)is equal to one of the invariant sets of the GIFS.The conclusion of this case is similar with the situation,which is the fractal square with any three sides of[0,1]~2 square.For four connected vertices,a GIFS is constructed by defining four label maps directly,which is similar with the four disconnected vertices are.