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November  2021, 41(11): 5243-5269. doi: 10.3934/dcds.2021075

The graph of the logistic map is a tower

 1 Department of Mathematics, Howard University, Washington, DC 20059, USA 2 Institute for Physical Science and Technology and the Departments of Mathematics and Physics, University of Maryland, College Park, MD 20742, USA

* Corresponding author: Roberto De Leo

In memory of Todd A. Drumm (1961-2020) and Tien-Yien Li (1945-2020).

Received  November 2020 Published  November 2021 Early access  April 2021

Fund Project: This material is based upon work partially supported by the National Science Foundation under Grant No. 1832126

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node $A$ to node $B$ if, using arbitrary small perturbations, a trajectory starting from any point of $A$ can be steered to any point of $B$. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node $A$ to node $B$, the unstable manifold of some periodic orbit in $A$ contains points that eventually map onto $B$. For special parameter values, this tower has infinitely many nodes.

Citation: Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075
References:
 [1] K. Alligood and J. Yorke, Accessible saddles on fractal basin boundaries, Ergodic Theory and Dynamical Systems, 12 (1992), 377-400.  doi: 10.1017/S0143385700006842.  Google Scholar [2] G. Birkhoff, Sur quelques courbes fermées remarquables, Bulletin de la Société mathématique de France, 60 (1932), 1-26.  Google Scholar [3] A. Blokh, The "spectral" decomposition for one-dimensional maps, in Dynamics Reported, Springer, 1995, 1-59.  Google Scholar [4] A. Blokh and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, in Annales Scientifiques de l'Ecole Normale Supérieure, 24 (1991), 545-573. doi: 10.24033/asens.1636.  Google Scholar [5] R. Bowen, $\omega$-limit sets for axiom A diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.  Google Scholar [6] J. Buescu, Exotic Attractors: From Liapunov Stability to Riddled Basins, 153, Birkhäuser, 2012. doi: 10.1007/978-3-0348-7421-2.  Google Scholar [7] X. Chen and P. Polácik, Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, Journal of Dynamics and Differential Equations, 7 (1995), 73-107.  doi: 10.1007/BF02218815.  Google Scholar [8] C. Conley, On a generalization of the morse index, in Ordinary Differential Equations, Elsevier, 1972, 27-33.  Google Scholar [9] C. Conley, Isolated Invariant Sets and the Morse Index, 38, American Mathematical Soc., 1978.  Google Scholar [10] R. De Leo and J. Yorke, Infinite towers in the graph of a dynamical system, Nonlinear Dynamics (to appear) Google Scholar [11] W. de Melo and S. van Strien, A structure theorem in one dimensional dynamics, Annals of Mathematics, 129 (1989), 519-546.  doi: 10.2307/1971516.  Google Scholar [12] W. de Melo and S. van Strien, One-Dimensional Dynamics, 25, Springer Science & Business Media, 1993. Available from: URL http://www2.imperial.ac.uk/ svanstri/Files/demelo-strien.pdf. doi: 10.1007/978-3-642-78043-1.  Google Scholar [13] M. Feigenbaum, Quantitative universality for a class of nonlinear transformations, Journal of Statistical Physics, 19 (1978), 25-52.  doi: 10.1007/BF01020332.  Google Scholar [14] J. Graczyk and G. Swiatek, Generic hyperbolicity in the logistic family, Annals of Mathematics, 1-52. doi: 10.2307/2951831.  Google Scholar [15] C. Grebogi, E. Ott and J. Yorke, Chaotic attractors in crisis, Physical Review Letters, 48 (1982), 1507. doi: 10.1103/PhysRevLett.48.1507.  Google Scholar [16] C. Grebogi, E. Ott and J. Yorke, Basin boundary metamorphoses: Changes in accessible boundary orbits, Nuclear Physics B-Proceedings Supplements, 2 (1987), 281-300.  doi: 10.1016/0167-2789(87)90078-9.  Google Scholar [17] J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Communications in Mathematical Physics, 70 (1979), 133-160.  doi: 10.1007/BF01982351.  Google Scholar [18] J. Hale, L. Magalhães and W. Oliva, Dynamics in Infinite Dimensions, 47, Springer Science & Business Media, 2006. Google Scholar [19] M. Hirsch, H. Smith and X. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, Journal of Dynamics and Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar [20] P. Holmes and D. Whitley, Bifurcations of one-and two-dimensional maps, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 311 (1984), 43-102.  doi: 10.1098/rsta.1984.0020.  Google Scholar [21] M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergodic Theory and Dynamical Systems, 11 (1991), 709-729.  doi: 10.1017/S014338570000643X.  Google Scholar [22] M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Communications in Mathematical Physics, 81 (1981), 39-88.  doi: 10.1007/BF01941800.  Google Scholar [23] L. Jonker and D. Rand, Bifurcations in one dimension, Inventiones Mathematicae, 62 (1980), 347-365.  doi: 10.1007/BF01394248.  Google Scholar [24] M. Lyubich, Dynamics of quadratic polynomials, I-II, Acta Mathematica, 178 (1997), 185-297.  doi: 10.1007/BF02392694.  Google Scholar [25] M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Mathematics, 1-78. doi: 10.2307/3597183.  Google Scholar [26] J. Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315.  doi: 10.1016/0022-0396(88)90157-X.  Google Scholar [27] J. Milnor, On the concept of attractor, in The Theory of Chaotic Attractors, Springer, 1985,243-264. doi: 10.1007/BF01212280.  Google Scholar [28] D. Norton, The Conley decomposition theorem for maps: a metric approach, Rikkyo Daigaku Sugaku Zasshi, 44 (1995), 151-173.   Google Scholar [29] D. Norton, The fundamental theorem of dynamical systems, Commentationes Mathematicae Universitatis Carolinae, 36 (1995), 585-597.   Google Scholar [30] M. Patrão, Morse decomposition of semiflows on topological spaces, Journal of Dynamics and Differential Equations, 19 (2007), 181-198.  doi: 10.1007/s10884-006-9033-2.  Google Scholar [31] K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, 1987. doi: 10.1007/978-3-642-72833-4.  Google Scholar [32] A. Sharkovsky, S. Kolyada, A. Sivak and V. Fedorenko, Dynamics of One-Dimensional Maps, 407, Springer Science & Business Media, 1997. doi: 10.1007/978-94-015-8897-3.  Google Scholar [33] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM Journal on Applied Mathematics, 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar [34] S. Smale, On gradient dynamical systems, Annals of Mathematics, 199-206. doi: 10.2307/1970311.  Google Scholar [35] S. Smale, Differentiable dynamical systems, Bulletin of the American mathematical Society, 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar [36] S. Smale and R. Williams, The qualitative analysis of a difference equation of population growth, Journal of Mathematical Biology, 3 (1976), 1-4.  doi: 10.1007/BF00307853.  Google Scholar [37] S. van Strien, On the bifurcations creating horseshoes, in Dynamical Systems and Turbulence, Warwick 1980, Springer, 1981,316-351.  Google Scholar [38] F. Wilson and J. Yorke, Lyapunov functions and isolating blocks, Journal of Differential Equations, 13 (1973), 106-123.  doi: 10.1016/0022-0396(73)90034-X.  Google Scholar

show all references

References:
 [1] K. Alligood and J. Yorke, Accessible saddles on fractal basin boundaries, Ergodic Theory and Dynamical Systems, 12 (1992), 377-400.  doi: 10.1017/S0143385700006842.  Google Scholar [2] G. Birkhoff, Sur quelques courbes fermées remarquables, Bulletin de la Société mathématique de France, 60 (1932), 1-26.  Google Scholar [3] A. Blokh, The "spectral" decomposition for one-dimensional maps, in Dynamics Reported, Springer, 1995, 1-59.  Google Scholar [4] A. Blokh and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, in Annales Scientifiques de l'Ecole Normale Supérieure, 24 (1991), 545-573. doi: 10.24033/asens.1636.  Google Scholar [5] R. Bowen, $\omega$-limit sets for axiom A diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.  Google Scholar [6] J. Buescu, Exotic Attractors: From Liapunov Stability to Riddled Basins, 153, Birkhäuser, 2012. doi: 10.1007/978-3-0348-7421-2.  Google Scholar [7] X. Chen and P. Polácik, Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, Journal of Dynamics and Differential Equations, 7 (1995), 73-107.  doi: 10.1007/BF02218815.  Google Scholar [8] C. Conley, On a generalization of the morse index, in Ordinary Differential Equations, Elsevier, 1972, 27-33.  Google Scholar [9] C. Conley, Isolated Invariant Sets and the Morse Index, 38, American Mathematical Soc., 1978.  Google Scholar [10] R. De Leo and J. Yorke, Infinite towers in the graph of a dynamical system, Nonlinear Dynamics (to appear) Google Scholar [11] W. de Melo and S. van Strien, A structure theorem in one dimensional dynamics, Annals of Mathematics, 129 (1989), 519-546.  doi: 10.2307/1971516.  Google Scholar [12] W. de Melo and S. van Strien, One-Dimensional Dynamics, 25, Springer Science & Business Media, 1993. Available from: URL http://www2.imperial.ac.uk/ svanstri/Files/demelo-strien.pdf. doi: 10.1007/978-3-642-78043-1.  Google Scholar [13] M. Feigenbaum, Quantitative universality for a class of nonlinear transformations, Journal of Statistical Physics, 19 (1978), 25-52.  doi: 10.1007/BF01020332.  Google Scholar [14] J. Graczyk and G. Swiatek, Generic hyperbolicity in the logistic family, Annals of Mathematics, 1-52. doi: 10.2307/2951831.  Google Scholar [15] C. Grebogi, E. Ott and J. Yorke, Chaotic attractors in crisis, Physical Review Letters, 48 (1982), 1507. doi: 10.1103/PhysRevLett.48.1507.  Google Scholar [16] C. Grebogi, E. Ott and J. Yorke, Basin boundary metamorphoses: Changes in accessible boundary orbits, Nuclear Physics B-Proceedings Supplements, 2 (1987), 281-300.  doi: 10.1016/0167-2789(87)90078-9.  Google Scholar [17] J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Communications in Mathematical Physics, 70 (1979), 133-160.  doi: 10.1007/BF01982351.  Google Scholar [18] J. Hale, L. Magalhães and W. Oliva, Dynamics in Infinite Dimensions, 47, Springer Science & Business Media, 2006. Google Scholar [19] M. Hirsch, H. Smith and X. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, Journal of Dynamics and Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar [20] P. Holmes and D. Whitley, Bifurcations of one-and two-dimensional maps, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 311 (1984), 43-102.  doi: 10.1098/rsta.1984.0020.  Google Scholar [21] M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergodic Theory and Dynamical Systems, 11 (1991), 709-729.  doi: 10.1017/S014338570000643X.  Google Scholar [22] M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Communications in Mathematical Physics, 81 (1981), 39-88.  doi: 10.1007/BF01941800.  Google Scholar [23] L. Jonker and D. Rand, Bifurcations in one dimension, Inventiones Mathematicae, 62 (1980), 347-365.  doi: 10.1007/BF01394248.  Google Scholar [24] M. Lyubich, Dynamics of quadratic polynomials, I-II, Acta Mathematica, 178 (1997), 185-297.  doi: 10.1007/BF02392694.  Google Scholar [25] M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Mathematics, 1-78. doi: 10.2307/3597183.  Google Scholar [26] J. Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315.  doi: 10.1016/0022-0396(88)90157-X.  Google Scholar [27] J. Milnor, On the concept of attractor, in The Theory of Chaotic Attractors, Springer, 1985,243-264. doi: 10.1007/BF01212280.  Google Scholar [28] D. Norton, The Conley decomposition theorem for maps: a metric approach, Rikkyo Daigaku Sugaku Zasshi, 44 (1995), 151-173.   Google Scholar [29] D. Norton, The fundamental theorem of dynamical systems, Commentationes Mathematicae Universitatis Carolinae, 36 (1995), 585-597.   Google Scholar [30] M. Patrão, Morse decomposition of semiflows on topological spaces, Journal of Dynamics and Differential Equations, 19 (2007), 181-198.  doi: 10.1007/s10884-006-9033-2.  Google Scholar [31] K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, 1987. doi: 10.1007/978-3-642-72833-4.  Google Scholar [32] A. Sharkovsky, S. Kolyada, A. Sivak and V. Fedorenko, Dynamics of One-Dimensional Maps, 407, Springer Science & Business Media, 1997. doi: 10.1007/978-94-015-8897-3.  Google Scholar [33] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM Journal on Applied Mathematics, 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar [34] S. Smale, On gradient dynamical systems, Annals of Mathematics, 199-206. doi: 10.2307/1970311.  Google Scholar [35] S. Smale, Differentiable dynamical systems, Bulletin of the American mathematical Society, 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar [36] S. Smale and R. Williams, The qualitative analysis of a difference equation of population growth, Journal of Mathematical Biology, 3 (1976), 1-4.  doi: 10.1007/BF00307853.  Google Scholar [37] S. van Strien, On the bifurcations creating horseshoes, in Dynamical Systems and Turbulence, Warwick 1980, Springer, 1981,316-351.  Google Scholar [38] F. Wilson and J. Yorke, Lyapunov functions and isolating blocks, Journal of Differential Equations, 13 (1973), 106-123.  doi: 10.1016/0022-0396(73)90034-X.  Google Scholar
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
] 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 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
. 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 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
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
. 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 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
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
]). 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 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 $\ell_\mu(x)$
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
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