Non-transitive points and porosity

T. K. Subrahmonian Moothathu¹

¹ School of Mathematics and Statistics University of Hyderabad Hyderabad 500046, India

Colloquium Mathematicum, Tome 133 (2013) no. 1, pp. 99-114

Résumé

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 $\sigma $-porosity, and in particular we show that the set of non-transitive points is $\sigma $-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 $\sigma $-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.

Détail

DOI : 10.4064/cm133-1-7

Keywords: establish fairly general class topologically transitive dynamical systems set non transitive points small rate transitivity high notion smallness consider here sigma porosity particular set non transitive points sigma porous subshift factor transitive subshift finite type tent map result extends finite to one factor systems family piecewise monotonic transitive interval maps set non transitive points sigma polynomially porous indicate similar methods sufficient conditions set non recurrent points set distal pairs dynamical system small