We establish that for a fairly general class of topologically transitive dynamical systems, the set of non-transitive points is very small when the rate of transitivity is very high. The notion of smallness that we consider here is that of σ-porosity, and in particular we show that the set of non-transitive points is σ-porous for any subshift that is a factor of a transitive subshift of finite type, and for the tent map of [0,1]. The result extends to some finite-to-one factor systems. We also show that for a family of piecewise monotonic transitive interval maps, the set of non-transitive points is σ-polynomially porous. We indicate how similar methods can be used to give sufficient conditions for the set of non-recurrent points and the set of distal pairs of a dynamical system to be very small.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-1-7,
author = {T. K. Subrahmonian Moothathu},
title = {Non-transitive points and porosity},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {99-114},
zbl = {1297.54073},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-1-7}
}
T. K. Subrahmonian Moothathu. Non-transitive points and porosity. Colloquium Mathematicae, Tome 131 (2013) pp. 99-114. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-1-7/