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Rotation numbers, periodic points and topological entropy of a class of endomorphisms of the circle

In this thesis we consider the dynamics of a class of endomorphisms of the circle, we denote this class of functions by Λ. Following the work of Milnor and Thurston [6] we develop a kneading theory for these functions, which enables us to calculate the entropy of our maps and to show that entropy is continuous. 
 
We then show that any map, f e Λ, with positive entropy is topologically semi-conjugate to a piecewise linear map, F. The map F is determined by two real numbers, the topological entropy of f and the twist number of f, both of which can be calculated from the kneading matrix. Using the fact that the rotation intervals of f and F are the same, we give a method of calculating this interval from the twist number and entropy of f. 
 
The final chapter is motivated by a theorem of Sarkovskii [9] and concerns universal properties of the periodic points.