Hausdorff Dimension Sierpinski Carpet

The determination of the.

Hausdorff dimension sierpinski carpet.

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. X t is an ε cover of ε 0 we define the hausdorff dimension of x by sup δ. The hausdorff dimension of the carpet is log 8 log 3 1 8928. The hausdorff dimension of general sierpiński carpets volume 96 curt mcmullen.

Hausdorff dimension exact value hausdorff dimension approx name illustration remarks calculated. Fractal hausdorff dimension and measure sierpinski carpet project partially supported by the science foundation of guangdong province. From then on some further problems related to the sierpinski carpet k t d are proposed and considered by lots of authors. Which is called fl quasi horizontal 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. Base diag n 1 m 1 with digits in d thesetk t d called as the sierpinski carpet was first studied by c. Xi hausdorff dimension of generalized sierpinski carpet 155 and dc to get a line segment. 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.

The hausdorff dimension of r let x be a metric space. Which is called fi quasi vertical segment. 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. The point of intersection of fi quasi vertical segment and fl quasi horizontal segment is.

A collection of sets x t is an ε cover of x if x uγ xt and diam x ε for all i. The δ dίmensional hausdorff measure of x is given by μ δ x supinf 2 diam xi δ. This is a preview of subscription content log in to check access. Sierpiński demonstrated that his carpet is a universal plane curve.

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. Sierpinski carpets 3 2. Bedford 1 independently to determine its hausdorff and box counting dimensions. Similarly we connect the points of definite proportion fl in ad and bc to get a line segment.

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.