Farey ağacının sınırı üzerindeki ölçüler

In this work, we consider trees, their boundaries and some special Borel measures on the boundary of trees. Especially we work on Farey Tree. Farey Tree is a rooted binary planar tree whose vertices are described by all rational numbers in the unit interval. It is the first left branch of the Stern-Brocot Tree. A topology on the boundary of the tree is induced by a metric, it is generated by open balls. Some natural Borel measures appear on the boundary of the Farey tree. The Minkowski measure is the unique measure on the boundary which is invariant under the full action of the automorphism group of the tree on its boundary. Denjoy's measures are a straightforward generalization of the Minkowski measure. Continued fraction map is defined on the unit interval which is a generalization of the Gauss map. The formal definition of this map is given in Chapter 5. In this thesis, we consider an even larger generalization of the continued fraction maps defined by using the tree structure. Our aim in this chapter is to find the invariant measures under these special continued fraction maps. Our claim is f(y)=1/y satisfies the density of an invariant measure under these special members of maps and we proved it. At the final chapter, we have a study that is not very compatible with the overall thesis. We associate a power series to continued fractions such that this series has some proprieties, such as convergence and continuity.