Linear Fractional Recurrences: Periodicities and Integrability TeX Export

@article{citeulike:6008337, abstract = {We consider k-step recurrences of the form \$z\_{n+k} = A(z)/B(z)\$, where A and B are linear functions of \$z\_n, z\_{n+1}, ..., z\_{n+k-1}\$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form \$A(z)/B(z) = (a +z\_{n+1} + z\_{n+2} + ... + z\_{n+k-1})/z\_n\$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.}, archivePrefix = {arXiv}, author = {Bedford, Eric and Kim, Kyounghee}, citeulike-article-id = {6008337}, citeulike-linkout-0 = {http://arxiv.org/abs/0910.4409}, citeulike-linkout-1 = {http://arxiv.org/pdf/0910.4409}, day = {22}, eprint = {0910.4409}, keywords = {dynamical\_systems, linear\_systems}, month = {Oct}, posted-at = {2009-11-01 10:00:07}, priority = {2}, title = {Linear Fractional Recurrences: Periodicities and Integrability}, url = {http://arxiv.org/abs/0910.4409}, year = {2009} }

TY - JOUR ID - citeulike:6008337 L3 - citeulike-article-id:6008337 N2 - We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals. TI - Linear Fractional Recurrences: Periodicities and Integrability KW - dynamical_systems KW - linear_systems AU - Bedford, Eric AU - Kim, Kyounghee PY - 2009/Oct/22/ UR - http://arxiv.org/abs/0910.4409 ER -