Shrinking targets and eventually always hitting points for interval maps

On Eventually Expanding Maps of the Interval

In this paper we conjecture that the piecewise linear map f(x) = px1I[0,1/p)(x)+ (sx − s/p)1I[1/p,1](x), p > 1, 0 < s < 1 which has an expanding, onto branch and a contracting branch is eventually piecewise expanding. We give a partial proof of the conjecture, in particular for values of p and s such that d− ln(p(1−s)+s) ln s e 6 = d− ln p ln s e.

Always and Eventually in Object Requirements

Object oriented models and methods encompass a set of techniques which have been, and will continue to be, applied in the successful production of complex software systems. The methods are based on the simple concepts of abstraction, encapsulation, classification and polymorphism. The formal verification of logical properties of such models is difficult to integrate into the traditional operati...

Invariant Measures for Interval Maps with Critical Points and Singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.

Shrinking Targets for Regularization of Logistic Regression

Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the central limit theorem and a vector-valued almost sure invariance principle. We also obtain results on decay of correlations.