# Generic invariant measures for iterated systems of interval homeomorphisms

@article{Czernous2019GenericIM, title={Generic invariant measures for iterated systems of interval homeomorphisms}, author={Wojciech Czernous and Tomasz Z. Szarek}, journal={Archiv der Mathematik}, year={2019}, volume={114}, pages={445-455} }

It is well known that iterated function systems generated by orientation preserving homeomorphisms of the unit interval with positive Lyapunov exponents at its ends admit a unique invariant measure on (0, 1) provided their action is minimal. With the additional requirement of continuous differentiability of maps on a fixed neighbourhood of $$\{0,1\}$$ { 0 , 1 } , we present a metric in the space of such systems which renders it complete. Using then a classical argument (and an alternative…

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## References

SHOWING 1-10 OF 22 REFERENCES

Random interval homeomorphisms

- Mathematics
- 2012

We investigate homeomorphisms of a compact interval, applied randomly. We consider this system as a skew product with the two-sided Bernoulli shift in the base. If on the open interval there is a…

Thick attractors of step skew products

- Mathematics
- 2010

A diffeomorphism is said to have a thick attractor provided that its Milnor attractor has positive but not full Lebesgue measure. We prove that there exists an open set in the space of boundary…

Random interval diffeomorphisms

- Mathematics
- 2016

We consider a class of step skew product systems of interval diffeo-morphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present…

Comparison of various concepts of a random attractor:¶ A case study

- Mathematics
- 2002

Abstract. Several non-equivalent definitions of an attractor of a random dynamical system have been proposed in the literature. We consider a rather simple special case: random dynamical systems with…

Attractors for random dynamical systems

- Mathematics
- 1994

SummaryA criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant…

Non-hyperbolic Iterated Function Systems: attractors and stationary measures

- Mathematics
- 2016

We consider iterated function systems $\mathrm{IFS}(T_1,\dots,T_k)$ consisting of continuous self maps of a compact metric space $X$. We introduce the subset $S_{\mathrm{t}}$ of {\emph{weakly…

Skew product attractors and concavity

- Mathematics
- 2012

We propose an approach to the attractors of skew products that tries to avoid unnecessary structures on the base space and rejects the assumption on the invariance of an attractor. When nonivertible…

Stability of the Markov operator and synchronization of Markovian random products

- Mathematics
- 2017

We study Markovian random products on a large class of "m-dimensional" connected compact metric spaces (including products of closed intervals and trees). We introduce a splitting condition,…

Invariant measures for Markov processes arising from iterated function systems with place-dependent

- Mathematics
- 1988

On considere un processus de Markov en temps discret sur un espace metrique localement compact obtenu par iteration aleatoire des cartes de Lipschitz w 1 , w 2 , ..., w n

Schwarzian Derivatives and Cylinder Maps

- Mathematics
- 2000

We describe the way in which the sign of the Schwarzian derivative for a family of dieomorphisms of the interval I aects the dynamics of an associated many-to-one skew product map of the cylinder…