On the piecewise linear perturbations of the doubling map

par Kuntal Banerjee (Presidency University, Kolkata)

vendredi 27 oct. 2023, 10:00 → 11:00 Europe/Paris

207 (Bat 1R2)

The family of piecewise linear perturbations of the doubling map (PLPDM) is defined as follows:
$$f_{a,b}=(2x+a+\frac{b}{2}S(x))\mathrm{mod}1\text{for }x\in \mathbb{T}=\mathbb{R}/\mathbb{Z}$$ 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.