# Invariant densities for C¹ maps

Studia Mathematica (1996)

- Volume: 120, Issue: 1, page 83-88
- ISSN: 0039-3223

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topQuas, Anthony. "Invariant densities for C¹ maps." Studia Mathematica 120.1 (1996): 83-88. <http://eudml.org/doc/216323>.

@article{Quas1996,

abstract = {We consider the set of $C^1$ expanding maps of the circle which have a unique absolutely continuous invariant probability measure whose density is unbounded, and show that this set is dense in the space of $C^1$ expanding maps with the $C^1$ topology. This is in contrast with results for $C^2$ or $C^\{1+ε\}$ maps, where the invariant densities can be shown to be continuous.},

author = {Quas, Anthony},

journal = {Studia Mathematica},

keywords = {cocycle; expanding map; invariant density; absolutely continuous invariant measure; invariant measures; expanding mappings},

language = {eng},

number = {1},

pages = {83-88},

title = {Invariant densities for C¹ maps},

url = {http://eudml.org/doc/216323},

volume = {120},

year = {1996},

}

TY - JOUR

AU - Quas, Anthony

TI - Invariant densities for C¹ maps

JO - Studia Mathematica

PY - 1996

VL - 120

IS - 1

SP - 83

EP - 88

AB - We consider the set of $C^1$ expanding maps of the circle which have a unique absolutely continuous invariant probability measure whose density is unbounded, and show that this set is dense in the space of $C^1$ expanding maps with the $C^1$ topology. This is in contrast with results for $C^2$ or $C^{1+ε}$ maps, where the invariant densities can be shown to be continuous.

LA - eng

KW - cocycle; expanding map; invariant density; absolutely continuous invariant measure; invariant measures; expanding mappings

UR - http://eudml.org/doc/216323

ER -

## References

top- [1] P. Góra and B. Schmitt, Un exemple de transformation dilatante et ${C}^{1}$ par morceaux de l’intervalle, sans probabilité absolument continue invariante, Ergodic Theory Dynam. Systems 9 (1989), 101-113. Zbl0672.58023
- [2] G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 2nd ed., Oxford Univ. Press, Oxford, 1992. Zbl0759.60002
- [3] K. Krzyżewski, A remark on expanding mappings, Colloq. Math. 41 (1979), 291-295. Zbl0446.58009
- [4] R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, New York, 1988.
- [5] A. N. Quas, Non-ergodicity for ${C}^{1}$ expanding maps and g-measures, Ergodic Theory Dynam. Systems 16 (1996), 1-13.
- [6] A. N. Quas, A ${C}^{1}$ expanding map of the circle which is not weak-mixing, Israel J. Math. 93 (1996), 359-372. Zbl0862.28015

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