Dimensions of compact invariant sets of some expanding maps

In this dissertation, we study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski carpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by considering a general Sierpinski carpet represented by a shift of finite type. Applying results of Ledrappier, Young, and Shin, we study the Hausdorff dimension of such a general Sierpinski carpet for the case when there is a saturated compensation function. We give some conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.