Invisible tricorns in real slices of rational maps

Figure 2.  For every parameter in the shaded region, the two free critical points of $ N_a $ are complex conjugate. The upper red component is denoted by $ \mathcal{U} $

Figure 1.  A part of the parameter plane of the family $ \mathcal{N}_4^* $ that contains a Tricorn component (enclosed by white curves) with two visible and one invisible boundary arcs. The white region at the bottom represents the complement of $ \mathcal{U} $

Figure 10.  Left: A Tricorn component (enclosed by white parabolic arcs) with an invisible parabolic arc. Right: A blow-up of the parameter plane near the invisible parabolic arc (on the left) shows an invisible Tricorn component (enclosed by white parabolic arcs)

Figure 11.  Left: A tongue component of period $ 2 $ with an invisible co-root parabolic arc $ \mathcal{C}_2 $. The parabolic arcs $ \mathcal{C}_1 $ and $ \mathcal{C}_3 $ contain bare regions. Right: A bounded Tricorn component with all three co-root parabolic arcs invisible

Figure 9.  The $ N_a^{\circ 2n} $-orbit of $ c_a $ exits through the gate and lands in $ W_a $, which is contained in the basin $ \mathcal{B}_1 $

Figure 3.  The main cardioid of the "baby Mandelbrot set" (in black) is a Mandelbrot component

Figure 5.  Subdivision rule that defines the sequence of critically finite branched covers $ f_{n}: \mathbb{S}^2\to \mathbb{S}^2 $ where $ n>1 $. Black edges $ \alpha, \beta, \gamma, \delta $ terminate at $ \infty $ in both the domain (top) and the range (bottom). Light blue edges in the domain represent preimage edges of $ f_{n} $ that are not edges in the range. Both domain and range graphs are symmetric about the horizontal and vertical coordinate axes.

Figure 4.  The dynamical plane of the map in $ \mathcal{N}_4^* $ corresponding to $ n = 3 $ from Proposition 5.3. The basins of $ 1 $ and $ -1 $ are colored red and green. The non-fixed critical points are in a period 6 cycle indicated by the white arrows

Figure 6.  Left: The characteristic Fatou component of a parameter belonging to a Tricorn component. The three dynamical co-roots, two of which are visible and one invisible, are marked. Right: The characteristic Fatou component of a parameter belonging to a Tricorn component. The three dynamical co-roots, each of which is invisible, are marked

Figure 7.  Left: A blow-up of the dynamical plane (of a parameter on a parabolic arc of a Tricorn component) around a visible characteristic parabolic point. Here, the characteristic parabolic point is on the boundary of $ \mathcal{B}^{\mathrm{imm}}_{-1} $. Right: A blow-up of the dynamical plane (of a parameter on a parabolic arc of a Tricorn component) around an invisible characteristic parabolic point. Here, the characteristic point is in the accumulation set of the pre-periodic Fatou components of $ \mathcal{B}_1 $ (and of $ \mathcal{B}_{-1} $), but not on the boundary of any single component thereof. Consequently, there is a 'Julia path' in the repelling cylinder connecting $ \partial U_1^+ $ and $ \partial U_1^- $

Figure 8.  A blow-up of the dynamical plane of a near-parabolic parameter showing the eggbeater dynamics "near" an invisible parabolic point

Figure 12.  Left: The dynamical plane of a parameter on the co-root arc $ \mathcal{C}_2 $ of a tongue of period $ 2 $. The only dynamical co-root on the boundary of $ U_1 $ is $ p_2(U_1) $ (which is also the characteristic parabolic point for parameters on $ \mathcal{C}_2 $), and it is invisible. The dynamical roots $ p_1(U_1) $ and $ p_3(U_1) $ on $ \partial U_1 $ are visible. Right: The dynamical plane of a parameter on a co-root arc $ \mathcal{C}_2 $ of a bounded Tricorn component. For this parameter, each dynamical co-root $ p_k(U_1) $ is invisible. In particular, the characteristic parabolic point $ p_2(U_1) $ is invisible (marked in black)