Circle maps with flat spots and unequal critical exponents at the boundaries have their parameter space partitioned into two regions by a phase transition boundary determined solely by the exponents.
and Palmisano, L., Ergodic properties of bimodal circle endomorphisms satisfying a Diophantine condition , Ergod
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A Phase Transition for Circle Maps with a Flat Spot and Different Critical Exponents
Circle maps with flat spots and unequal critical exponents at the boundaries have their parameter space partitioned into two regions by a phase transition boundary determined solely by the exponents.