Functional Analysis behind a Family of Multidimensional Continued Fractions: Part I
Abstract
Triangle partition maps form a family that includes many, if not most, wellknown multidimensional continued fraction algorithms. This paper begins the exploration of the functional analysis behind the transfer operator of each of these maps. We show that triangle partition maps give rise to two classes of transfer operators and present theorems regarding the origin of these classes; we also present related theorems on the form of transfer operators arising from compositions of triangle partition maps. In the next paper, Part II, we will find eigenfunctions of eigenvalue 1 for transfer operators associated with select triangle partition maps on specified Banach spaces and then proceed to prove that the transfer operators, viewed as acting on onedimensional families of Hilbert spaces, associated with select triangle partition maps are nuclear of trace class zero. We will finish in part II by deriving GaussKuzmin distributions associated with select triangle partition maps.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.01589
 Bibcode:
 2017arXiv170301589A
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Number Theory
 EPrint:
 Original preprint split into two parts. This is part I.. To appear in Publicationes Mathematicae Debrecen. Small errors also corrected