Towards some conjectures by Arnol’d

Abstract In this work, we focus on the study of the components of semilevel sets of pseudoperiodic functions. Specifically, we provide partial proofs of a conjecture proposed by V.I. Arnol’d in 1991, concerning the boundedness of such components. In pursuit of a complete answer to this conjecture, we also present an estimate for the length of a path connecting points within components that satisfy a key property. This study can be considered a sequel to a work published by the author in 2024. Nevertheless, we have endeavored to make the text as self-contained as possible by maintaining consistent notation throughout.