Geodesic
The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations
Mathematica Bohemica, Tome 122 (1997) no. 1, pp. 37-55
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In this paper piecewise monotonic maps $T [0,1]\to[0,1]$ 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 $(R(Q),T)$.
In this paper piecewise monotonic maps $T [0,1]\to[0,1]$ 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 $(R(Q),T)$.
DOI : 10.21136/MB.1997.126187
Classification : 37D99, 37E99, 54H20, 58F03, 58F15, 58f30
Keywords: piecewise monotonic map; nonwandering set; topologically transitive subset 
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Raith, Peter. The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations. Mathematica Bohemica, Tome 122 (1997) no. 1, pp. 37-55. doi: 10.21136/MB.1997.126187

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