Invariant densities for C¹ maps

Anthony Quas

Studia Mathematica (1996)

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

Abstract

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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.

How to cite

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Quas, 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

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  1. [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. [2] G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 2nd ed., Oxford Univ. Press, Oxford, 1992. Zbl0759.60002
  3. [3] K. Krzyżewski, A remark on expanding mappings, Colloq. Math. 41 (1979), 291-295. Zbl0446.58009
  4. [4] R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, New York, 1988. 
  5. [5] A. N. Quas, Non-ergodicity for C 1 expanding maps and g-measures, Ergodic Theory Dynam. Systems 16 (1996), 1-13. 
  6. [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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