Hausdorff Dimension Sierpinski Carpet

X t is an ε cover of ε 0 we define the hausdorff dimension of x by sup δ.

Hausdorff dimension sierpinski carpet.

The hausdorff dimension of the carpet is log 8 log 3 1 8928. From then on some further problems related to the sierpinski carpet k t d are proposed and considered by lots of authors. Base diag n 1 m 1 with digits in d thesetk t d called as the sierpinski carpet was first studied by c. The δ dίmensional hausdorff measure of x is given by μ δ x supinf 2 diam xi δ.

The hausdorff dimension of general sierpiński carpets volume 96 curt mcmullen. Sierpiński demonstrated that his carpet is a universal plane curve. 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. Similarly we connect the points of definite proportion fl in ad and bc to get a line segment.

When you would just fill all the holes in the sierpinski triangle except for the big one in the middle you would get exactly the same hausdorff dimension log2 3 1 585 even though this reduced triangle is at least from my current perspective obviously less rough than the original sierpinski mesh. Which is called fl quasi horizontal segment. Hausdorff dimension exact value hausdorff dimension approx name illustration remarks calculated. Xi hausdorff dimension of generalized sierpinski carpet 155 and dc to get a line segment.

The determination of the. Fractal hausdorff dimension and measure sierpinski carpet project partially supported by the science foundation of guangdong province. A collection of sets x t is an ε cover of x if x uγ xt and diam x ε for all i. The hausdorff dimension of r let x be a metric space.

Which is called fi quasi vertical segment. 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. 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. This is a preview of subscription content log in to check access.

μ δ x 00. Bedford 1 independently to determine its hausdorff and box counting dimensions. 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 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.

The point of intersection of fi quasi vertical segment and fl quasi horizontal segment is.