Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors

Figure 1.  Dynamics of $ \Gamma_{\mu} .$

Figure 2.  The smoothness domains for a map in $ \mathbb{F} .$

Figure 3.  The set $ \mathcal{R}_1 .$

Figure 4.  The set of parameters $ \mathcal{P}_1 .$

Figure 5.  The set $ \mathcal{R}_{a, b} .$

Figure 6.  The set of parameters $ \mathcal{P}_2 .$

Figure 7.  The set $ \mathcal{R}_{a, b} .$

Figure 8.  The sets $ \Delta_{a, b} $ and $ \Pi_{a, b} .$

Figure 9.  The iterates of $ \Delta_0:$ encircled in blue, the triangle $ \Delta_0 ;$ encircled in a dashed blue line, the set $ \Delta_4 .$

Figure 10.  The set of parameters $ \mathcal{P}_{\Delta} .$

Figure 11.  The sets of parameters $ \mathcal{P}_{\Delta}^{\prime} $ (dark grey) and $ \mathcal{P}_{\Delta} $ (pale grey).

Figure 12.  The iterates of $ \Pi_0 :$ encircled in green, the triangle $ \Pi_0 ;$ encircled in a dashed green line, the set $ \Pi_4 .$

Figure 13.  The set of parameters $ \mathcal{P}_{\Pi} .$

Figure 14.  (a) Filled in black, the set $ \mathcal{P}_3 ;$ encircled in a dashed black line, the set $ H_{\Delta}(\mathcal{P}_3) .$ (b) Filled in black, the set $ \mathcal{P}_3 ;$ encircled in a dashed black line, the set $ H_{\Pi}(\mathcal{P}_3) .$

Figure 15.  The relative position between $\gamma_0$ and $\eta_n.$