Linear Fractional Recurrences: Periodicities and Integrability

Eric Bedford[1]; Kyounghee Kim[2]

  • [1] Department of Mathematics, Indiana University, Bloomington, IN 47405 USA
  • [2] Department of Mathematics, Florida State University, Tallahassee, FL 32306 USA

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: S2, page 33-56
  • ISSN: 0240-2963

Abstract

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Linear fractional recurrences are given as z n + k = A ( z ) / B ( z ) , where A ( z ) and B ( z ) are linear functions of z n , z n + 1 , ⋯ , z n + k - 1 . In this article we consider two questions about these recurrences: (1) Find A ( z ) and B ( z ) such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each k there are k -step linear fractional recurrences which are periodic of period 4 k . The second theorem shows that the Lyness process, A ( z ) = a + z n + 1 + z n + 2 + ⋯ + z n + k - 1 and B ( z ) = z n + 1 has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.

How to cite

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Bedford, Eric, and Kim, Kyounghee. "Linear Fractional Recurrences: Periodicities and Integrability." Annales de la faculté des sciences de Toulouse Mathématiques 20.S2 (2011): 33-56. <http://eudml.org/doc/219703>.

@article{Bedford2011,
abstract = {Linear fractional recurrences are given as $z_\{n+k\} = A(z)/B(z)$, where $A(z)$ and $B(z)$ are linear functions of $z_n, z_\{n+1\}, \dots , z_\{n+k-1\}$. In this article we consider two questions about these recurrences: (1) Find $A(z)$ and $B(z)$ such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each $k$ there are $k$-step linear fractional recurrences which are periodic of period $4k$. The second theorem shows that the Lyness process, $A(z) = a+ z_\{n+1\} + z_\{n+2\} + \cdots + z_\{n+k-1\}$ and $B(z) = z_\{n+1\}$ has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.},
affiliation = {Department of Mathematics, Indiana University, Bloomington, IN 47405 USA; Department of Mathematics, Florida State University, Tallahassee, FL 32306 USA},
author = {Bedford, Eric, Kim, Kyounghee},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {birational maps; non-periodicity; linear fractional recurrences},
language = {eng},
month = {4},
number = {S2},
pages = {33-56},
publisher = {Université Paul Sabatier, Toulouse},
title = {Linear Fractional Recurrences: Periodicities and Integrability},
url = {http://eudml.org/doc/219703},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Bedford, Eric
AU - Kim, Kyounghee
TI - Linear Fractional Recurrences: Periodicities and Integrability
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - S2
SP - 33
EP - 56
AB - Linear fractional recurrences are given as $z_{n+k} = A(z)/B(z)$, where $A(z)$ and $B(z)$ are linear functions of $z_n, z_{n+1}, \dots , z_{n+k-1}$. In this article we consider two questions about these recurrences: (1) Find $A(z)$ and $B(z)$ such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each $k$ there are $k$-step linear fractional recurrences which are periodic of period $4k$. The second theorem shows that the Lyness process, $A(z) = a+ z_{n+1} + z_{n+2} + \cdots + z_{n+k-1}$ and $B(z) = z_{n+1}$ has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.
LA - eng
KW - birational maps; non-periodicity; linear fractional recurrences
UR - http://eudml.org/doc/219703
ER -

References

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