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Abstract
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.
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Authors (2)
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Wojciech Czernous
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- Category:
- Articles
- Type:
- artykuły w czasopismach
- Published in:
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ARCHIV DER MATHEMATIK
no. 114,
pages 445 - 455,
ISSN: 0003-889X - Language:
- English
- Publication year:
- 2020
- Bibliographic description:
- Czernous W., Szarek T.: Generic invariant measures for iterated systems of interval homeomorphisms// ARCHIV DER MATHEMATIK -Vol. 114, (2020), s.445-455
- DOI:
- Digital Object Identifier (open in new tab) 10.1007/s00013-019-01405-7
- Verified by:
- Gdańsk University of Technology
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