The Hausdorff Dimension Of The Graph Of Fractal Functions

This thesis mainly includes two parts:The first part calculates the Hausdorff dimensions of the graphs of a class of self-affine fractal interpolation functions.The second part calculates the Hausdorff dimensions of the graphs of some classical Weierstrass functions.The formula for the box dimensions of the graphs of FIFs was given in 1980s by Barns-ley,Hardin,Massopust and etc.However,there are no good results for their Hausdorff dimen-sions.Usually we can only obtain the upper and lower bound of Hausdorff dimension.In this paper,we regard the graphs of some FIFs as attractors of some dynamical systems and obtain their Hausdorff dimensions by using Keller's method published in 2017.It is one of the most important problems in fractal geometry to study the fractal dimensions of the graphs of Weierstrass functions.While their box dimensions have been solved,their Haus-dorff dimensions are quite complex.It's conjectured that for all Weierstrass functions,their box dimension and Hausdorff dimension are equal.In 2017,Weixiao Shen proved this conjecture is true for cosine type Weierstrass functions under some conditions.We know that if pushforward measure is absolutely continuous with respect to the Lebesgue measure,then the conjecture is true.Shen treated their graphs as repellors of some dynamical systems and proved the absolute continuity of pushforward measure.In this thesis,inspired by Shen,we calculate the Hausdorff dimensions of the graphs of a class of sine type Weierstrass functions.