The family of piecewise linear perturbations of the doubling map (PLPDM) is defined as follows:
$$ f_{a,b}= \left(2x+a+\frac{b}{2} S(x) \right) \mod 1 \quad {\text {for }} x\in \mathbb T=\mathbb R/\mathbb Z$$ where $S(x)$ is the piecewise linear (PL) approximation to $\sin 2\pi(x-\frac{1}{4})$. The parameter space of this family is $\mathcal P=\{(a,b):a\in \mathbb R, 0\le b<1\}$.
Define the union of the hyperbolic components as $$\mathcal{H}= \{ (a,b) \in [0,1]^2 : f_{a,b} \text{ has an attracting cycle}\}.$$
Tongues are defined as the components of $\mathcal{H}$ that touch the ceiling $b=1$. Any other component will be referred to as an Eye.
We show the existence of the tongues and finiteness of attracting cycles for any map in this family associated to a parameter belonging to a hyperbolic component of $\mathcal H$. We also discuss the transitivity of the maps in this family, the possible nature of wandering intervals and a new type of bifurfaction in this family. Some experimental proof of eyes in the parameter space will be shown as well.
This is a joint work with Anubrato Bhattacharyya with inputs from Alexandre Dezotti.
