Quadratic maps without asymptotic measure

@article{Hofbauer1990QuadraticMW,
  title={Quadratic maps without asymptotic measure},
  author={Franz Hofbauer and Gerhard Keller},
  journal={Communications in Mathematical Physics},
  year={1990},
  volume={127},
  pages={319-337}
}
An interval map is said to have an asymptotic measure if the time averages of the iterates of Lebesgue measure converge weakly. We construct quadratic maps which have no asymptotic measure. We also find examples of quadratic maps which have an asymptotic measure with very unexpected properties, e.g. a map with the point mass on an unstable fix point as asymptotic measure. The key to our construction is a new characterization of kneading sequences. 

Topological conditions for the existence of invariant measures for unimodal maps

  • H. Bruin
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1994
Abstract We present a class of S-unimodal maps having an invariant measure which is absolutely continuous with respect to Lebesgue measure. This measure can often be proved to be finite. We give an

Exact $^{∞}$ covering maps of the circle without (weak) limit measure

We construct C 1 maps T on the interval and on the circle which are Lebesgue-exact preserving an absolutely continuous infinite measure µ , such that for any probability measure the sequence (n 1 P n

Critical covering maps without absolutely continuous invariant probability measure

We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than \begin{document}$ 1 $\end{document} . By directly specifying the combinatorics of the

Statistical properties of unimodal maps: the quadratic family

We prove that almost every nonregular real quadratic map is ColletEckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps

The Existence of ?nite Invariant Measures, Applications to Real 1-dimensional Dynamics

A general construction for ?nite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of f n () will imply

Almost every real quadratic map is either regular or stochastic

In this paper we complete a program to study measurable dynamics in the real quadratic family. Our goal was to prove that almost any real quadratic map Pc : z t- x2 + c, c c [-2,1/4], has either an

The existence of σ-finite invariant measures, Applications to real onedimensional dynamics. Front for the Math

A general construction for σ−finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of fn ∗ (λ) will

Quadratic Maps with Maximal Oscillation

Let (f t )o≤t≤1 denote the family of quadratic maps f t(x) = 2t(1- x 2) - 1 on [-1, 1]. An important aspect of the asymptotics of interates of a map f t is the behaviour of mass distributions along

Emergence for diffeomorphisms with nonzero Lyapunov exponents

We consider the set of points with high pointwise emergence for C diffeomorphisms preserving a hyperbolic measure. We find a lower bound on the Hausdorff dimension of this set in terms of unstable

Countable limit sets of unimodal maps

We show that for any positive integer n, there exists a unimodal map whose turning point has a countable ω-limit set of depth n. We also show that the quadratic map F(x) = 1 − 2x2 admits countable
...

References

SHOWING 1-10 OF 32 REFERENCES

Positive Liapunov exponents and absolute continuity for maps of the interval

Abstract We give a sufficient condition for a unimodal map of the interval to have an invariant measure absolutely continuous with respect to the Lebesgue measure. Apart from some weak regularity

Exponents, attractors and Hopf decompositions for interval maps

  • G. Keller
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1990
Abstract Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following: (1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals

Some properties of absolutely continuous invariant measures on an interval

Abstract 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

Lifting measures to Markov extensions

Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we

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

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

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

We consider a class of piecewise monotonically increasing functionsf on the unit intervalI. We want to determine the measures with maximal entropy for these transformations. In part I we construct a

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II

The results about measures with maximal entropy, which are proved in [3], are extended to the following more general class of transformations on the unit intervalI : I=∪i=1/nJi, theJi are disjoint

Iterates of Maps on an Interval

Piecewise monotone functions.- Well-behaved piecewise monotone functions.- Property R and negative schwarzian derivatives.- The iterates of functions in S.- Reductions.- Getting rid of homtervals.-

On iterated maps of the interval

Introduction. Mappings from an interval to itself provide the simplest possible examples of smooth dynamical systems. Such mappings have been widely studied in recent years since they occur in quite

Absolutely continuous invariant measures forC2 unimodal maps satisfying the Collet-Eckmann conditions

SummaryIn this paper we show that unimodal mappingsf∶[0, 1]→[0, 1] have absolutely continuous measures of positive entropy if these maps areC2 and satisfy the so-called Collet-Eckmann conditions. No