• DOI:10.1007/s00013-019-01405-7
• Corpus ID: 173990410

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}
}

• W. Czernous, T. Szarek
• Published 31 May 2019
• Mathematics
• Archiv der Mathematik

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 uniqueness proof), we show that almost singular invariant measures are admitted by systems lying densely in the space. This allows us to construct a residual set of systems with unique singular stationary distribution. Dichotomy between singular and absolutely continuous unique measures is assured by taking a subspace of systems with absolutely continuous maps; the closure of this subspace is where the residual set is found.