TY - JOUR

T1 - ALTERNATING N-EXPANSIONS

AU - Dajani, Karma

AU - Langeveld, Niels

PY - 2022/8/7

Y1 - 2022/8/7

N2 - We introduce a family of maps generating continued fractions where the digit 1 in the numerator is replaced cyclically by some given non-negative integers (N_1, . . . , N_m). We prove the convergence of the given algorithm, and study the underlying dynamical system generating such expansions. We prove the existence of a unique absolutely continuous invariant ergodic measure. In special cases, we are able to build the natural extension and give an explicit expression of the invariant measure. For these cases, we formulate a Doeblin-Lenstra type theorem. For other cases we have a more implicit expression that we conjecture gives the invariant density. This conjecture is supported by simulations. For the simulations we use a method that gives us a smooth approximation in every iteration.

AB - We introduce a family of maps generating continued fractions where the digit 1 in the numerator is replaced cyclically by some given non-negative integers (N_1, . . . , N_m). We prove the convergence of the given algorithm, and study the underlying dynamical system generating such expansions. We prove the existence of a unique absolutely continuous invariant ergodic measure. In special cases, we are able to build the natural extension and give an explicit expression of the invariant measure. For these cases, we formulate a Doeblin-Lenstra type theorem. For other cases we have a more implicit expression that we conjecture gives the invariant density. This conjecture is supported by simulations. For the simulations we use a method that gives us a smooth approximation in every iteration.

KW - N -continued fractions,

KW - invariant density

KW - ergodicity

KW - natural extensions.

M3 - Article

VL - 2022

JO - INTEGERS: Electronic Journal of Combinatorial Number Theory

JF - INTEGERS: Electronic Journal of Combinatorial Number Theory

SN - 1867-0652

IS - 22

M1 - A65

ER -