Presentation functions and scaling function theory for circle maps

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Abstract
Considering first return maps, a most natural renormalisation group fixed point is determined. From it a simple presentation function is constructed, immediately leading to the thermodynamics of critical rotation. The rotation number is encoded in the topological action of the presentation function and the algebraic singularity of critically in that function's derivatives at its fixed points. Any such presentation function determines a circle map dynamics of that rotation number and index of criticality. These functions are naturally parametrised by a trajectory scaling function. The requirement that the dynamics be smooth leads to a prescription for the calculation of the scaling function and hence the dynamics. The theory is highly constrained and suffers in finite-order approximation from the extra constraint of commutativity, which however can be overcome.

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