Determine which branch of the stationary-measure dichotomy occurs

Determine general methods for deciding whether a given stationary measure for a uniformly hyperbolic, strongly Diophantine family of Möbius transformations satisfies alternative (I) or alternative (II) of the paper’s $L^q$-spectrum dichotomy.

Background

The main theorem proves that the LqL^q spectrum of a stationary measure associated with a uniformly hyperbolic and strongly Diophantine family of elements of SL2(R)\mathrm{SL}_2(\mathbb R) has exactly one of two possible forms. In alternative (I), it equals the expected pressure value for every q>1q>1; in alternative (II), it eventually becomes linear, of the form αq\alpha q, after a threshold q0q_0.

The paper explicitly states that the theorem itself does not provide a way to determine which alternative applies to a specified stationary measure and that no general decision method is obtained. Developing such methods would distinguish ordinary behavior from the singular behavior caused by complicated overlaps.

References

Theorem \ref{the_main_theorem_Lq_dim_Mobius_IFS} gives no information to determine which (I) or (II) occurs for a given stationary measure and we can't obtain general methods to do this in this paper. The author thinks that it is much meaningful to give such methods.

— 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)