The graph of the logistic map is a tower

Figure 1.  Bifurcations diagram and sample graphs of the logistic map. For each value of $ \mu $, the attracting set is painted in shades of gray, depending on the density of the attractor, repelling periodic orbits in green, and repelling Cantor sets in red. The "black dots" that are visible within the diagram are low-period periodic points. They signal the presence of bifurcation cascades within some window. The fact that some of them keep close to the line $ x = c $ is a reflection of Singer's Theorem: when the attractor is a periodic orbit, $ c $ belongs to its immediate basin. For selected values of $ \mu $ we also show, below the diagram, the graph of the corresponding logistic map

Figure 2.  Towers of nodes in the period-3 window of the logistic map. The period-3 window $ W $ starts at $ \mu\simeq3.8284 $ and ends at $ \mu\simeq3.8568 $. At every $ \mu $ within $ W $, an invariant Cantor set node $ N $ (in red in figure) arises, depending continuously on $ \mu $. This is exactly the Cantor set discussed by Smale and Williams in [36] at $ \mu = 3.83 $. The widest white interval within the Cantor set is the $ J_1(N) $ interval of the period-3 trapping region $ {{\mathcal T}}(N) $ running throughout $ W $. The periodic endpoint $ p_1(N) $ of $ J_1(N) $ is the top endpoint, the bottom one is $ q_1 = 1-p_1(N) $. The blue Cantor set node $ N' $ visible in figure is the node of a regular trapping region $ {{\mathcal T}}(N') $ nested in $ {{\mathcal T}}(N) $. As in Fig. 1, below the diagram we show the graph of the logistic map for selected values of $ \mu $

Figure 3.  Flip and regular trapping regions associated to a periodic orbit node. The fixed point $ p_1 $ is a node for $ 1<\mu<\mu_M $ (see Eq. 3). Each node has its own $ p_1 $, $ q_1 $ and $ J_1 $. It is attracting for $ \mu<3 $. For $ \mu<2 $, the trapping region associated with it consists of a single interval $ {J}_1 = [p_1, q_1] $, where $ q_1 = 1-p_1 $. As $ \mu $ increases past the super-stable value $ \mu = 2 $, the trapping region becomes flip and consists of two intervals $ {J}_1 = [q_1, p_1] $ and $ {J}_2 = [p_1, q_2] $, with $ \ell_\mu(q_2) = q_1 $. The flip trapping region ends when $ p_1 $ hits the chaotic attractor at $ \mu_M $ (Eq. 3). The region $ 3.4\leq\mu\leq3.6 $ is shown in greater detail in Fig. 6

Figure 4.  A trapping region associated to a Cantor set node. This window is a blowup of a region in Fig. 2. The red region is a Cantor set (for each $ \mu $ in the window). The Cantor set is a node. The interval $ J_1 = [q_1, p_1] $ is a trapping region of $ \ell_\mu^3 $ for each $ \mu $ in the window. Each node has a trapping region for $ \ell_\mu $ and $ J_1 $ is the piece of the trapping region that contains the critical point. For each $ \mu $ in the interior of the window, the point $ p_1 $ belongs to a repelling period-3 orbit within the red Cantor set and $ \ell_\mu(q_1) = p_1 $. There is a $ \mu $ Within $ J_1 $ arises a bifurcation diagram qualitatively identical to the full one. The Cantor set node in blue within the period-3 window of the diagram inside $ J_1 $ is the analog of the red Cantor set within the main period-3 window. Fig. 7 shows the same region but with more detail

Figure 5.  An example of graph. (LEFT) Dynamics induced on the 2-torus by the gradient vector field of the height function. In this case the Lyapunov function is the height function itself, some level set of which is shaded in white. In blue are shown the heteroclinic trajectories joining the critical point (which are exactly the invariant sets of this dynamical system). (CENTER) The graph of the dynamical system on the left. In this case it is a 4-levels tower. (RIGHT) An infinite tower

Figure 6.  Flip trapping regions of the logistic map. This picture is a blowup of Fig. 3. It shows at the three parameter values $ \mu_{60} = 3.43, \mu_{61} = 3.5, \mu_{62} = 3.56 $, all flip cyclic trapping regions of the logistic map with a repelling orbit at their boundary. At $ \mu = \mu_{60} $, there is a single flip trapping region $ {{\mathcal T}} = \{{J}_1, {J}_2\} $, where $ {J}_1 = [q_1, p_1] $ and $ {J}_2 = [p_1, q_2] $. The point $ p_1 $ is the flip fixed point, $ q_1 = 1-p_1 $ and $ \ell_\mu(q_2) = q_1 $. A description for the other two values of $ \mu $ is given in the text below Def. 3.7

Figure 7.  Examples of regular and flip cyclic trapping regions. In this detail of the period-3 window, the curve of points denoted by $ p_1 $ is a period-3 repellor. It separates the red Cantor set from the white basin of the attractor. The curve of points $ q_1 = 1-p_1 $ separates the Cantor set from the basin. A description for the marked values of $ \mu $ is given in the text below Def. 3.7

Figure 8.  A regular cyclic trapping region for the logistic map. The bifurcation diagram for the logistic map has a period-3 window in parameter space starting at $ \mu_0 = 1+\sqrt{8}\simeq3.828 $, where a pair of an attracting and a repelling period-3 orbits arise, and ending at $ \mu_1\simeq3.857 $, where the unstable periodic orbit collides with the attractor (i.e. there is a crises [15]). Here we show what is happening at one of the intermediate parameter values, $ \mu = 3.854 $. There are intervals $ {J}_i $ with endpoints $ q_i, p_i $, $ i = 1, 2, 3 $, shown in red, which, together, form a period-3 regular cyclic trapping region for $ \ell_\mu $ (Def. 3.5). The arrows show how the endpoints map under $ \ell_\mu. $ The intervals are chosen so that $ p_1\mapsto p_2\mapsto p_3\mapsto p_1 $ is an unstable period-3 orbit and $ \ell_\mu^3(q_i) = p_i $ for $ i = 1, 2, 3 $. This construction gives rise to a cyclic trapping region because, for this value of $ \mu $ and this choice of the endpoints, $ \ell_\mu^3({J}_i)\subset {J}_i $. The picture also includes the graph of $ \ell_\mu^3(x) $ and $ \ell_\mu(x) $

Figure 9.  Nested flip cyclic trapping regions for the logistic map. On the left panel, we show an interval $ [p_1, q_1] $ that is the $ {J}_1 $ interval of a period-$ k $ cyclic trapping region $ {{\mathcal T}} $ for $ F = \ell_\mu^k $. Inside $ {J}_1 $, we show a flip cyclic trapping region (Def. 3.5) $ {{\mathcal T}}' = \{{J}'_1, {J}'_2\} $, where $ {J}_1' = [q_1', p_1'] $ and $ {J}'_2 = [p'_1, q'_2] $. On the right panel, again inside $ {J}_1 $, we show a flip cyclic trapping region $ {{\mathcal T}}" = \{{J}_1", {J}_2", {J}_3", {J}"_4\} $ nested in $ {{\mathcal T}}' $, where $ {J}"_1 = [p_1", q_1"] $, $ {J}"_2 = [p"_2, q"_2] $, $ {J}"_3 = [q"_3, p"_1] $ and $ {J}"_4 = [q"_4, p"_2] $. The arrows in both panels show how the endpoints map under $ F $. On the left, the periodic orbit at the boundary of the cyclic trapping region is the fixed point $ p'_1 $, on the right is the period-2 orbit $ p"_1\mapsto p"_2\mapsto p"_1 $. Notice that $ F^2({J}'_1)\subset {J}'_1 $ and $ F^4({J}"_1)\subset {J}_1" $. The picture also includes the graphs of $ F(x) $, $ F^2(x) $ and $ F^4(x) $. The actual value used in these pictures is $ \mu = \mu_{FM} $; at this value, there is an infinite sequence of flip cyclic trapping regions nested one into the other