# Most expanding maps have no absolutely continuous invariant measure

Studia Mathematica (1999)

- Volume: 134, Issue: 1, page 69-78
- ISSN: 0039-3223

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topQuas, Anthony. "Most expanding maps have no absolutely continuous invariant measure." Studia Mathematica 134.1 (1999): 69-78. <http://eudml.org/doc/216623>.

@article{Quas1999,

abstract = {We consider the topological category of various subsets of the set of expanding maps from a manifold to itself, and show in particular that a generic $C^1$ expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for $C^2$ or $C^\{1+ε\}$ expanding maps, for which it is known that there is always a unique absolutely continuous invariant probability measure.},

author = {Quas, Anthony},

journal = {Studia Mathematica},

keywords = {$C^1$ expanding map; Ruelle-Perron-Frobenius operator; expanding maps of the circle; invariant probability measures},

language = {eng},

number = {1},

pages = {69-78},

title = {Most expanding maps have no absolutely continuous invariant measure},

url = {http://eudml.org/doc/216623},

volume = {134},

year = {1999},

}

TY - JOUR

AU - Quas, Anthony

TI - Most expanding maps have no absolutely continuous invariant measure

JO - Studia Mathematica

PY - 1999

VL - 134

IS - 1

SP - 69

EP - 78

AB - We consider the topological category of various subsets of the set of expanding maps from a manifold to itself, and show in particular that a generic $C^1$ expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for $C^2$ or $C^{1+ε}$ expanding maps, for which it is known that there is always a unique absolutely continuous invariant probability measure.

LA - eng

KW - $C^1$ expanding map; Ruelle-Perron-Frobenius operator; expanding maps of the circle; invariant probability measures

UR - http://eudml.org/doc/216623

ER -

## References

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- [3] P. Góra et 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
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- [6] K. Krzyżewski, A remark on expanding mappings, Colloq. Math. 41 (1979), 291-295. Zbl0446.58009
- [7] K. Krzyżewski and W. Szlenk, On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83-92. Zbl0176.00901
- [8] M. R. Palmer, W. Parry and P. Walters, Large sets of endomorphisms and of g-measures, in: The Structure of Attractors in Dynamical Systems, Lecture Notes in Math. 668, Springer, Berlin, 1978, 191-210.
- [9] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque 187-188 (1990). Zbl0726.58003
- [10] A. N. Quas, Invariant densities for ${C}^{1}$ maps, Studia Math. 120 (1996), 83-88. Zbl0858.58030
- [11] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc. 214 (1975), 375-387. Zbl0331.28013

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