Hausdorff Dimension Sierpinski Carpet

The δ dίmensional hausdorff measure of x is given by μ δ x supinf 2 diam xi δ.

Hausdorff dimension sierpinski carpet. The authors are very grateful to huojun ruan for providing a method to construct the sierpiński carpets not julia sets with hausdorff dimension one theorem d to xiaoguang wang yongcheng yin and jinsong zeng for offering a proof of lemma 5 2 we would also like to thank arnaud chéritat kevin pilgrim feliks przytycki weiyuan qiu yongcheng yin and anna zdunik for helpful discussions. Which is called fl quasi horizontal segment. Xi hausdorff dimension of generalized sierpinski carpet 155 and dc to get a line segment. X t is an ε cover of ε 0 we define the hausdorff dimension of x by sup δ.

Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. The determination of the. The point of intersection of fi quasi vertical segment and fl quasi horizontal segment is. Sierpiński demonstrated that his carpet is a universal plane curve.

The sierpinski carpet is a compact subset of the plane with lebesgue covering dimension 1 and every subset of the plane with these properties is homeomorphic to some subset of the sierpiński carpet. Mcmullen 6 and t. From then on some further problems related to the sierpinski carpet k t d are proposed and considered by lots of authors. Up to now the accurate value of hausdorff measure of some self similar sets with hausdorff dimension no more than 1 has been obtained such as middle three cantor set 1.

Similarly we connect the points of definite proportion fl in ad and bc to get a line segment. Which is called fi quasi vertical segment. μ δ x 00. The hausdorff dimension of r let x be a metric space.

In 2 the hausdorff measures of certain sierpinski carpets whose hausdorff dimension equal to 1 are obtained with the help of the principle of mass distribution. This is a preview of subscription content log in to check access. Fractal hausdorff dimension and measure sierpinski carpet project partially supported by the science foundation of guangdong province. Hausdorff dimension exact value hausdorff dimension approx name illustration remarks calculated.

The hausdorff dimension of general sierpiński carpets volume 96 curt mcmullen. Bedford 1 independently to determine its hausdorff and box counting dimensions. Sierpinski carpets 3 2. A collection of sets x t is an ε cover of x if x uγ xt and diam x ε for all i.

The feigenbaum attractor see between arrows is the set of points generated by successive iterations of the logistic function for the critical parameter value where the period doubling is infinite this dimension is the same for any differentiable and.