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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  7. P. Raith, 10.1007/BF02808156, Israel J. Math. 80 (1992), 97-133. (1992) Zbl0768.28010MR1248929DOI10.1007/BF02808156
  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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