For a sufficiently well-behaved X, the Hausdorff dimension is the unique number d such that N( r) grows as 1/ r d as r approaches zero. When r is very small, N( r) grows polynomially with 1/ r. Consider the number N( r) of balls of radius at most r required to cover X completely. The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space. A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1.īut topological dimension is a very crude measure of the local size of a space (size near a point). This dimension is n if, in every covering of X by small open balls, there is at least one point where n+1 balls overlap. The topological dimension, also called Lebesgue covering dimension, explains why. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information.) The example of a space-filling curve shows that one can even take one real number into two both surjectively (so all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.Įvery space filling curve hits some points multiple times, and does not have a continuous inverse. The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. 4.4 Behaviour under unions and products.
![hausdorff dimension subshift angle doubling hausdorff dimension subshift angle doubling](https://www.wernerparts.com/media/wysiwyg/alothemes/static/cms/chart.gif)
4.3 Hausdorff dimensions and Frostman measures.4.2 Hausdorff dimension and Minkowski dimension.4.1 Hausdorff dimension and inductive dimension.