The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations

Peter Raith

Mathematica Bohemica (1997)

  • Volume: 122, Issue: 1, page 37-55
  • ISSN: 0862-7959

Abstract

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In this paper piecewise monotonic maps T [ 0 , 1 ] [ 0 , 1 ] are considered. Let Q be a finite union of open intervals, and consider the set R ( Q ) of all points whose orbits omit Q . The influence of small perturbations of the endpoints of the intervals in Q on the dynamical system ( R ( Q ) , T ) is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary small perturbations of Q . Furthermore it is shown that every sufficiently “big” maximal topologically transitive subset of a sufficiently small perturbation of ( R ( Q ) , T ) is “dominated” by a topologically transitive subset of ( R ( Q ) , T ) .

How to cite

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Raith, Peter. "The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations." Mathematica Bohemica 122.1 (1997): 37-55. <http://eudml.org/doc/248140>.

@article{Raith1997,
abstract = {In this paper piecewise monotonic maps $T [0,1]\rightarrow [0,1]$ are considered. Let $Q$ be a finite union of open intervals, and consider the set $R(Q)$ of all points whose orbits omit $Q$. The influence of small perturbations of the endpoints of the intervals in $Q$ on the dynamical system $(R(Q),T)$ is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary small perturbations of $Q$. Furthermore it is shown that every sufficiently “big” maximal topologically transitive subset of a sufficiently small perturbation of $(R(Q),T)$ is “dominated” by a topologically transitive subset of $(R(Q),T)$.},
author = {Raith, Peter},
journal = {Mathematica Bohemica},
keywords = {piecewise monotonic map; nonwandering set; topologically transitive subset; piecewise monotonic map; nonwandering set; topologically transitive subset},
language = {eng},
number = {1},
pages = {37-55},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations},
url = {http://eudml.org/doc/248140},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Raith, Peter
TI - The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations
JO - Mathematica Bohemica
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 122
IS - 1
SP - 37
EP - 55
AB - In this paper piecewise monotonic maps $T [0,1]\rightarrow [0,1]$ are considered. Let $Q$ be a finite union of open intervals, and consider the set $R(Q)$ of all points whose orbits omit $Q$. The influence of small perturbations of the endpoints of the intervals in $Q$ on the dynamical system $(R(Q),T)$ is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary small perturbations of $Q$. Furthermore it is shown that every sufficiently “big” maximal topologically transitive subset of a sufficiently small perturbation of $(R(Q),T)$ is “dominated” by a topologically transitive subset of $(R(Q),T)$.
LA - eng
KW - piecewise monotonic map; nonwandering set; topologically transitive subset; piecewise monotonic map; nonwandering set; topologically transitive subset
UR - http://eudml.org/doc/248140
ER -

References

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  4. M. Misiurewicz S. V. Shlyachkov, Entropy of piecewise continuous interval maps, European conference on iteration theory (ECIT 89), Batschuns, 1989 (Ch. Mira, N. Netzer, C. Simó, Gy. Targoński, eds.). World Scientific, Singapore, 1991, pp. 239-245. (1989) MR1184170
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  8. P. Raith, Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta Math. Univ. Comenian. 63 (1994), 39-53. (1994) Zbl0828.58014MR1342594
  9. P. Raith, The dynamics of piecewise monotonic maps under small perturbations, Preprint, Warwick, 1994. (1994) MR1627314
  10. M. Urbański, Hausdorff dimension of invariant sets for expanding maps of a circle, Ergodic Theory Dynam. Systems 6 (1986), 295-309. (1986) MR0857203
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  12. P. Walters, 10.1007/978-1-4612-5775-2, Graduate Texts in Mathematics 79, Springer, New York, 1982. (1982) Zbl0475.28009MR0648108DOI10.1007/978-1-4612-5775-2

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