# A generalization of Zeeman’s family

Fundamenta Mathematicae (1999)

• Volume: 162, Issue: 3, page 277-286
• ISSN: 0016-2736

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## Abstract

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E. C. Zeeman [2] described the behaviour of the iterates of the difference equation ${x}_{n+1}=R\left({x}_{n},{x}_{n-1},...,{x}_{n-k}\right)/Q\left({x}_{n},{x}_{n-1},...,{x}_{n-k}\right)$, n ≥ k, R,Q polynomials in the case $k=1,Q={x}_{n-1}$ and $R={x}_{n}+\alpha$, ${x}_{1},{x}_{2}$ positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.

## How to cite

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Sierakowski, Michał. "A generalization of Zeeman’s family." Fundamenta Mathematicae 162.3 (1999): 277-286. <http://eudml.org/doc/212424>.

@article{Sierakowski1999,
abstract = {E. C. Zeeman [2] described the behaviour of the iterates of the difference equation $x_\{n+1\} = R(x_n,x_\{n-1\},...,x_\{n-k\})/Q(x_n,x_\{n-1\},..., x_\{n-k\})$, n ≥ k, R,Q polynomials in the case $k = 1, Q = x_\{n-1\}$ and $R = x_n+α$, $x_1,x_2$ positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.},
author = {Sierakowski, Michał},
journal = {Fundamenta Mathematicae},
keywords = {difference equation; critical line; totally invariant set},
language = {eng},
number = {3},
pages = {277-286},
title = {A generalization of Zeeman’s family},
url = {http://eudml.org/doc/212424},
volume = {162},
year = {1999},
}

TY - JOUR
AU - Sierakowski, Michał
TI - A generalization of Zeeman’s family
JO - Fundamenta Mathematicae
PY - 1999
VL - 162
IS - 3
SP - 277
EP - 286
AB - E. C. Zeeman [2] described the behaviour of the iterates of the difference equation $x_{n+1} = R(x_n,x_{n-1},...,x_{n-k})/Q(x_n,x_{n-1},..., x_{n-k})$, n ≥ k, R,Q polynomials in the case $k = 1, Q = x_{n-1}$ and $R = x_n+α$, $x_1,x_2$ positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.
LA - eng
KW - difference equation; critical line; totally invariant set
UR - http://eudml.org/doc/212424
ER -

## References

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1. [1] F. Beukers and R. Cushman, Zeeman's monotonicity conjecture, J. Differential Equations 143 (1998), 191-200. Zbl0944.37026
2. [2] E. C. Zeeman, A geometric unfolding of a difference equation, J. Difference Equations Appl., to appear.
3. [3] E. C. Zeeman, Higher dimensional unfoldings of difference equations, lecture notes, ICTP Conference, Trieste, September 1998.

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