Abstract. Recently there has been a very intense development about the dimension theory of self-similar and self-affine fractals.

Abstract: In this talk, we consider the attractors of iterated function systems which are more general, and we want to estimate their dimension. The term Iterated Function System (IFS) on the d-dimensional space, means merely a finite list of contractive self-maps of the d-dimensional space. In this talk, we consider families of IFSs which are neither affine nor conformal in the d-dimensional space, where d is at least 2.  If we choose a closed ball B centered at the origin with a sufficiently large radius, then all of these maps send B into itself. If we take the union of all n-fold iterations of the mappings of the IFS applied on B, then these unions for different n, form a nested sequence of compact sets. Their intersection is the attractor of the IFS under consideration. We want to express the dimension of the attractor with other invariants like entropy and Lyapunov exponents for some measures naturally associated with the dynamical system or with the root of the so-called pressure formula. We can give effective lower estimates only in some special cases. However, we can give upper bound under very general conditions. Our upper bound estimates work even if the IFS under consideration consists of  C^1 and not only for C^{1+\epsilon} functions.  (Based on our joint results with De-Jun Feng)