Most expanding maps have no absolutely continuous invariant measure

Anthony Quas

Studia Mathematica (1999)

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

Abstract

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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.

How to cite

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Quas, 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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  1. [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Amer. Math. Soc., Providence, 1997. Zbl0882.28013
  2. [2] H. Bruin and J. Hawkins, Examples of expanding C 1 maps having no σ-finite measure equivalent to Lebesgue, preprint, 1996. Zbl0936.37001
  3. [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
  4. [4] J. M. Hawkins and C. E. Silva, Noninvertible transformations admitting no absolutely continuous σ-finite invariant measure, Proc. Amer. Math. Soc. 111 (1991), 455-463. Zbl0729.28012
  5. [5] K. Krzyżewski, On expanding mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 23-24. Zbl0208.06903
  6. [6] K. Krzyżewski, A remark on expanding mappings, Colloq. Math. 41 (1979), 291-295. Zbl0446.58009
  7. [7] K. Krzyżewski and W. Szlenk, On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83-92. Zbl0176.00901
  8. [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. [9] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque 187-188 (1990). Zbl0726.58003
  10. [10] A. N. Quas, Invariant densities for C 1 maps, Studia Math. 120 (1996), 83-88. Zbl0858.58030
  11. [11] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc. 214 (1975), 375-387. Zbl0331.28013

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