For Möbius iterated function systems satisfying the strongly Diophantine condition, the Lq spectrum of the stationary measure either equals the expected value for all q, or it becomes a linear function with slope below 1 for all large q.
Feng, Gibbs properties of self-conformal measures and the multifractal formalism, Ergodic Theory Dynam
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On the $L^q$ dimension of stationary measures for M\"{o}bius iterated function systems
For Möbius iterated function systems satisfying the strongly Diophantine condition, the Lq spectrum of the stationary measure either equals the expected value for all q, or it becomes a linear function with slope below 1 for all large q.