Topological dynamics of piecewise λ-affine maps
Let I be the unit interval [0,1) and -1 < λ < 1. Let f : I → R be a piecewise λ-affine map, that is, there are real numbers b_1;,…, b_u and a sequence of points 0 = c_1 < c_2 <…< c_{u-1} < c_u = 1 such that f(x)= λx + b_i, for every x in the interval [c_{i-1} , c_i). In the talk, we examine the class of maps f_ρ = f + ρ mod 1, where ρ is a real parameter. We prove that, for Lebesgue almost every real parameter rho, the map f_ρ is asymptotically periodic. More precisely, f_ρ has at most 2n periodic orbits and the ω-limit set of every x in the unit interval I is a periodic orbit. Our theorem is an extension of the result obtained for the much-studied case where λ is a positive constant and f is the continuous map x → λx . This is a joint work with Benito Pires and Rafael Rosales.