Generic invariant measures for iterated systems of interval homeomorphisms

Wojciech Czernous , Tomasz Szarek


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}, 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.
Author Wojciech Czernous (FMPI/IM)
Wojciech Czernous,,
- Institute of Mathematics
, Tomasz Szarek (FMPI/IM)
Tomasz Szarek,,
- Institute of Mathematics
Journal seriesArchiv der Mathematik, ISSN 0003-889X, e-ISSN 1420-8938, (N/A 70 pkt)
Issue year2020
Publication size in sheets0.50
Keywords in EnglishMarkov operators, semigroups of interval homeomorphisms, absolute continuity, singularity, minimal actions
ASJC Classification2600 General Mathematics
Languageen angielski
LicenseJournal (articles only); published final; Uznanie Autorstwa (CC-BY); with publication
Score (nominal)70
Score sourcejournalList
ScoreMinisterial score = 70.0, 20-07-2020, ArticleFromJournal
Publication indicators WoS Citations = 0.000; Scopus SNIP (Source Normalised Impact per Paper): 2018 = 0.692; WoS Impact Factor: 2018 = 0.498 (2) - 2018=0.556 (5)
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