Non-transitive points and porosity
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.