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

Mathematica Bohemica (1997)

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

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topRaith, 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 -

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