Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

@article{Jakobson1981AbsolutelyCI,
  title={Absolutely continuous invariant measures for one-parameter families of one-dimensional maps},
  author={Michael Jakobson},
  journal={Communications in Mathematical Physics},
  year={1981},
  volume={81},
  pages={39-88}
}
  • M. Jakobson
  • Published 1 September 1981
  • Mathematics
  • Communications in Mathematical Physics
Given a one-parameter familyfλ(x) of maps of the interval [0, 1], we consider the set of parameter values λ for whichfλ has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)fλ(x)=λf(x) where 0<λ≦4 andf(x) is a functionC3-near the quadratic mapx(1−x), and ii)fλ(x)=λf(x) (mod 1) wheref isC3,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1]. 
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References

SHOWING 1-10 OF 11 REFERENCES
Invariant measures for Markov maps of the interval
There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When
INVARIANT MEASURES AND EQUILIBRIUM STATES FOR SOME MAPPINGS WHICH EXPAND DISTANCES
For a certain collection of transformations T we define a Perron- Frobenius operator and prove a convergence theorem for the powers of the operator along the lines of the theorem D. Ruelle proved in
Some Dynamical Properties of Certain Differentiable Mappings of an Interval
This paper is a continuation of the paper [S]. We present here a sufficient condition for the existence of invariant measure (absolutely continuous with respect to Lebesgue measure) for
Sensitive dependence to initial conditions for one dimensional maps
This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The
Some properties of absolutely continuous invariant measures on an interval
We are interested in ergodic properties of absolutely continuous invariant measures of positive entropy for a map of an interval. We prove a Bernoulli property and a characterization by some
Absolutely continuous measures for certain maps of an interval
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Sufficient conditions are given such that a differentiable, non invertible, mapg:[0, 1]↦[0, 1] leaves invariant a measure absolutely continuous with respect to the Lebesgue measure. In particular,
F-expansions revisited
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