Contemporary Mathematics Scaling Functions for Degree 2 Circle Endomorphisms G. Cui, F. P. Gardiner and Y. Jiang 1 Abstract. We prove that a continuous function on the dual Cantor set is the scaling function of a uniformly symmetric circle endomorphism if and only if it satisfies a summation condition and compatibility condition. We use this result to establish an isomorphism between the space of continuous functions on the dual Cantor set satisfying these conditions and a Teichm¨ uller space. Introduction Scaling functions for Markov maps (see [8, 12] for the definition) have been used to describe fine-scale geometric structures of dynamical systems. For a Markov map one can construct a semi-conjugacy between the map and a shift map acting on a symbolic space of finite type. The scaling function, if it exists, is defined on a symbolic space dual to this space (see Section 1 for the definition). A natural problem is to determine which functions defined on a given dual symbolic space are the scaling functions of a Markov map. A circle endomorphism of degree two is a special Markov map with a stan- dard Markov partition. Using this partition, one obtains a semi-conjugacy between the one-sided full shift on two symbols and the circle endomorphism. If the endo- morphism is expanding and C 1+h , then it has a scaling function which is H¨older continuous on the dual symbolic space (see [8, Chapter 3]). However, the scaling function exists for a larger class of circle expanding endomorphism, namely, for uni- formly symmetric circle endomorphisms (see Section 1 for the definition). In this case the scaling function on the dual symbolic space is continuous only (see Theo- rem 1 in Section 1). However, not every continuous function on the dual symbolic space is the scaling function of an endomorphism. To be a scaling function, it must satisfy a trivial condition which we call the summation condition and a non-trivial convergence condition which we call the compatibility condition. Using Gibbs mea- sures, it is proved in [4] that these two conditions are necessary and sufficient for a H¨older continuous function defined on the dual symbolic space to be the scaling function of a C 1+h circle endomorphism. Since the existence of Gibbs measures depends on the H¨older continuity condition (or the bounded summable variation condition) (see [2, 4]), there seems little hope that the same method could apply to the more general situation where the circle endomorphism is uniformly symmetric. 1 Partially supported by grants from NNSF of China, NSF and PSC-CUNY 2000 Mathematics Subject Classification. Primary 58F23; Secondary 30C62. c 0000 (copyright holder) 1