Compute the threshold and slope in the exceptional spectrum

Determine the exact values of the threshold $q_0$ and slope $\alpha$ for a stationary measure belonging to alternative (II) of the $L^q$-spectrum dichotomy.

Background

In alternative (II), the paper establishes the existence of constants q0>1q_0>1 and 0<α<10<\alpha<1 such that the LqL^q spectrum agrees with the expected formula for 1<q<q01<q<q_0 and equals αq\alpha q for q≥q0q\geq q_0.

The paper explains that this exceptional behavior appears to result from complicated overlaps in the underlying Möbius iterated function system. Although the theorem guarantees the existence of α\alpha and q0q_0, the paper does not provide a procedure for calculating them or analyzing the overlaps sufficiently to recover their values.

References

The author thinks that, for a stationary measure $\nu$, even if one can know that $\nu$ is of the case (II), it is difficult to know the exact value of $\alpha$ or $q_0$. It seems that the case (II) occurs because of complicated overlaps of the IFS. Hence, to know the value of $\alpha$ or $q_0$, we have to study such overlaps in much detail and he does not know how to do that.

— On the $L^q$ dimension of stationary measures for Möbius iterated function systems  (2501.13729 - Usuki, 23 Jan 2025) in Remark following Theorem 1.4, Section 1.2 (the main theorem)