Non-transitive points and porosity

Volume 133 / 2013

T. K. Subrahmonian Moothathu Colloquium Mathematicum 133 (2013), 99-114 MSC: Primary 54H20, 26A16; Secondary 37B10, 37E05, 28A05. DOI: 10.4064/cm133-1-7


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.


  • T. K. Subrahmonian MoothathuSchool of Mathematics and Statistics
    University of Hyderabad
    Hyderabad 500046, India

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