Non-transitive points and porosity

T. K. Subrahmonian Moothathu. "Non-transitive points and porosity." Colloquium Mathematicae 133.1 (2013): 99-114. <http://eudml.org/doc/283853>.

@article{T2013,
abstract = {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.},
author = {T. K. Subrahmonian Moothathu},
journal = {Colloquium Mathematicae},
keywords = {porosity and -porosity; transitivity; subshifts; interval map; Lipschitz and Hölder continuity; recurrence; proximality},
language = {eng},
number = {1},
pages = {99-114},
title = {Non-transitive points and porosity},
url = {http://eudml.org/doc/283853},
volume = {133},
year = {2013},
}

TY - JOUR
AU - T. K. Subrahmonian Moothathu
TI - Non-transitive points and porosity
JO - Colloquium Mathematicae
PY - 2013
VL - 133
IS - 1
SP - 99
EP - 114
AB - 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.
LA - eng
KW - porosity and -porosity; transitivity; subshifts; interval map; Lipschitz and Hölder continuity; recurrence; proximality
UR - http://eudml.org/doc/283853
ER -