# A generalization of Zeeman’s family

Fundamenta Mathematicae (1999)

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

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topSierakowski, 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

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

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