| Title:
|
The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations (English) |
| Author:
|
Raith, Peter |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
122 |
| Issue:
|
1 |
| Year:
|
1997 |
| Pages:
|
37-55 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $(R(Q),T)$. (English) |
| Keyword:
|
piecewise monotonic map |
| Keyword:
|
nonwandering set |
| Keyword:
|
topologically transitive subset |
| MSC:
|
37D99 |
| MSC:
|
37E99 |
| MSC:
|
54H20 |
| MSC:
|
58F03 |
| MSC:
|
58F15 |
| MSC:
|
58f30 |
| idZBL:
|
Zbl 0896.58027 |
| idMR:
|
MR1446398 |
| DOI:
|
10.21136/MB.1997.126187 |
| . |
| Date available:
|
2009-09-24T21:22:33Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126187 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[7] P. Raith: Continuity of the Hausdorff dimension for piecewise monotonic maps.Israel J. Math. 80 (1992), 97-133. Zbl 0768.28010, MR 1248929, 10.1007/BF02808156 |
| Reference:
|
[8] P. Raith: Continuity of the Hausdorff dimension for invariant subsets of interval maps.Acta Math. Univ. Comenian. 63 (1994), 39-53. Zbl 0828.58014, MR 1342594 |
| Reference:
|
[9] P. Raith: The dynamics of piecewise monotonic maps under small perturbations.Preprint, Warwick, 1994. MR 1627314 |
| Reference:
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[10] M. Urbański: Hausdorff dimension of invariant sets for expanding maps of a circle.Ergodic Theory Dynam. Systems 6 (1986), 295-309. MR 0857203 |
| Reference:
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| Reference:
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| . |
