Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system （UIFS） on R^d. Denote by D the Hausdorff dimension, by H^D（E） the Hausdorff measu...Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system （UIFS） on R^d. Denote by D the Hausdorff dimension, by H^D（E） the Hausdorff measure and by diam（E） the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H^O（E） = diam（E）^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H^D（E） 〈 diam（E）^D, if c is small enough.展开更多
文摘Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system （UIFS） on R^d. Denote by D the Hausdorff dimension, by H^D（E） the Hausdorff measure and by diam（E） the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H^O（E） = diam（E）^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H^D（E） 〈 diam（E）^D, if c is small enough.
