ALTERNATING N-EXPANSIONS
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In: INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 2022, No. 22, A65, 07.08.2022.
Research output: Contribution to journal › Article › Research › peer-review
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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 -