| Title:
|
On the continuity of the pressure for monotonic mod one transformations (English) |
| Author:
|
Raith, Peter |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
41 |
| Issue:
|
1 |
| Year:
|
2000 |
| Pages:
|
61-78 |
| . |
| Category:
|
math |
| . |
| Summary:
|
If $f:[0,1]\to{\Bbb R}$ is strictly increasing and continuous define $T_fx=f(x)\, (\operatorname{mod} 1)$. A transformation $\tilde{T}:[0,1]\to [0,1]$ is called $\varepsilon$-close to $T_f$, if $\tilde{T}x=\tilde{f}(x)\, (\operatorname{mod} 1)$ for a strictly increasing and continuous function $\tilde{f}:[0,1]\to{\Bbb R}$ with $\|\tilde{f}-f\|_{\infty}<\varepsilon$. It is proved that the topological pressure $p(T_f,g)$ is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function $g:[0,1]\to{\Bbb R}$, if and only if $0$ is not periodic or $1$ is not periodic. Finally it is shown that the topological entropy is continuous, if $h_{\text{\rm top}}(T_f)>0$. (English) |
| Keyword:
|
mod one transformation |
| Keyword:
|
topological pressure |
| Keyword:
|
topological entropy |
| Keyword:
|
maximal measure |
| Keyword:
|
perturbation |
| MSC:
|
37B40 |
| MSC:
|
37D35 |
| MSC:
|
37E05 |
| MSC:
|
37E99 |
| MSC:
|
54H20 |
| idZBL:
|
Zbl 1034.37021 |
| idMR:
|
MR1756927 |
| . |
| Date available:
|
2009-01-08T18:58:33Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119141 |
| . |
| 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:
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| Reference:
|
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| . |
