Characterizing the RIFs that induce bounded composition operators on the Bergman space of the bidisc

In this short paper we will discuss recent advances on the problem of characterizing the boundedness of the composition operator acting on the Bergman spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">D</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> Aβ²(D²) whenever the self map Φ of the bidisc is induced by Rational Inner Functions. The problem stated here is submitted as part of the Problem List of the Young Researchers Workshop in Complex Analysis and Operator Theory “The Bench Math Session 2025”, organized in Jagiellonian University of Kraków at 10th–11th February 2025.