We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape $ (+-+) $ which was posed in [
Citation: |
Figure 1. The real moduli space $ \mathcal{M}_2(\Bbb{R}) $ as illustrated in [14,Figure 15]. The post-critical lines $ \sigma_1 = -6, 2 $ ($ \omega_1, \omega_2 $ denote the critical points of $ f $ here), the dotted lines $ {\rm{Per}}_1(\pm 1) $, the real symmetry locus $ \mathcal{S}(\Bbb{R}) $ and the partition into seven regions according to the various types of the dynamics induced on $ \hat{\Bbb{R}} $ are shown. The component of degree zero maps in $ \mathcal{M}_2(\Bbb{R})-\mathcal{S}(\Bbb{R}) $ is the union of monotonic, unimodal and bimodal regions that overlap only along the lines $ \sigma_1 = -6, 2 $
Figure 2. A colored version of Figure 1. The complement in $ \mathcal{M}_2(\Bbb{R}) $ of the symmetry locus admits three connected components corresponding to possible topological degrees of the restriction $ f\!\!\restriction_{\hat{\Bbb{R}}} $ of a quadratic rational map $ f $ with real coefficients. If the degree is $ \pm 2 $, the restriction is a covering map of entropy $ \log(2) $. The entropy behavior in the component of degree zero maps (in pink) is far more interesting
Figure 3. An entropy contour plot in the $ (+-+) $-bimodal region of the real moduli space (the $ (\sigma_1, \sigma_2) $-plane, cf. Figure 1) adapted from [10]. Here the colors blue, magenta, green, cyan, yellow and red correspond to the entropy being in intervals $ [0, 0.05) $, $ [0.05, 0.2) $, $ [0.2, 0.3) $, $ [0.3, 0.5) $, $ [0.5, 0.66) $ and $ [0.66, \log(2)\approx 0.7] $ respectively. The plot is generated utilizing the algorithm introduced in [1]; and black indicates the failure of that algorithm in calculating the entropy. The right vertical boundary line is the post-critical line $ \sigma_1 = -6 $ which intersects the lower skew boundary line $ {\rm{Per}}_1(1): \sigma_2 = 2\sigma_1-3 $ (both of them visible in Figure 1). For $ (+-+) $-bimodal maps below this line the Julia set is completely real and the real entropy is $ \log(2) $ [10,§4]. The real entropy tends to zero as we tend to the upper boundary which is part of the symmetry locus
Figure 4. An illustration of the real dynamics described in Proposition 3.2 in the case of $ q = 10, p = 3 $. Deployment of the post-critical set $ \left\{x_j\right\}_{j = 0}^{q-1} $ (in black) and the distinguished repelling $ q $-cycle $ \left\{\zeta_j\right\}_{j = 0}^{q-1} $ (in red) is drawn on the circle (at bottom) and on the real line (at top); compare with statement j of the proposition. The map $ f $ permutes them as $ x_j\mapsto x_{j+p} $ and $ \zeta_j\mapsto\zeta_{j+p} $. There is a cycle of open intervals between $ \zeta_j $'s and $ x_j $'s (in green) which lies in the immediate super-attracting basin of $ \left\{x_j\right\}_{j = 0}^{q-1} $. The repelling fixed point $ {\rm{i}} $ is at the center of the disk, while the repelling fixed point at $ -{\rm{i}} $ is the point at infinity in this representation. The Markov partition formed by intervals $ I_j = [x_j, x_{j+1}] $ is also visible in the picture
Figure 5. The dynamical $ w $-plane in the case of $ q = 8, p = 3 $. In the proof of Proposition 3.2, one first constructs a rational map $ g = g(w) $ which is conjugate to the desired $ f_{p/q}(z) $ via (17). The construction is by the means of "blowing up" a $ p/q $-rotation along a curve $ \gamma $. In this process, $ \gamma $ is slit open (here to the black ellipse) and a topological disk $ D $ (here the interior of the ellipse and in red) is then inserted. The endpoints $ c_0 $ and $ c_1 $ of $ \gamma $ turn out to be the critical points of the resulting rational map $ g $ and the center of the rotation a repelling fixed point. The post-critical set $ {\rm{P}}_g $ is the set of black points. The star-shaped curve $ \Gamma $ comes up in establishing property i
Figure 6. Dynamical plane of $ f_{p/q, t} $ for $ p/q = 1/3 $ and $ t = 0.45 $. Drawn as circles, not to scale, are fundamental domains for the local return map at the attractor (lower) and a repelling fixed point. Drawn as solid lines are the three lifts of the geodesics $ \gamma_{t, +} $ joining each element of the attracting 3-cycle (normalized here to be $ 0, 1, \infty $) to the repelling fixed point. The region between the dashed lines is the lift of an annular neighborhood of $ \gamma_{t, +} $ whose modulus tends to infinity as $ t \uparrow 1 $. This region projects to the quotient torus of the repelling fixed point, forcing its multiplier $ \mu_+(t) $ to tend to $ {\rm{e}}^{{2\pi {\rm{i}}}/{3}} $ as $ t \uparrow 1 $. Petersen's estimates utilized in the proof of Proposition 3.9 are in terms of the multipliers of repelling fixed points $ \lambda(t) $ for the Blaschke product of the return map on the attracting basin. The situation in the lower-half plane is symmetric, via reflection in the real axis
Figure 7. An illustration of real bitransitive hyperbolic components adapted from [14,Figure 17]. Compare the limit points with Remark 3.10
Figure 8. The portion of the compactification $ \overline{\mathcal{M}_2(\Bbb{R})} $ of $ \mathcal{M}_2(\Bbb{R}) $ which is located to the left of the post-critical line $ \sigma_1 = -6 $; compare with Figure 1. The boundary circle (12) (in thick black) and the ideal points on it (in black bold font) are visible. The lines $ {\rm{Per}}_1(\pm 1) $ are the loci where one of the real fixed points becomes parabolic (hence of multiplier $ +1 $ or $ -1 $); and the colored regions cut by them lie in the escape components. Between these two lines we have $ (+-+) $-bimodal maps and certain curves relevant to the proof of Theorem 1.1. Each entropy value $ h\in\left(h_3, h_q\right] $ is realized on the purple segments but not on the broken red curve $ L $, hence the disconnectedness of the level set $ h_\Bbb{R} = h $
Figure 9. A graph of the real entropy along the post-critical line $ \sigma_1 = -6 $ versus the coordinate $ \sigma_2 $. The topological entropy has been calculated via the algorithm developed in [2] for unimodal maps. The entropy is a decreasing function of $ \sigma_2 $; cf. Lemma 4.2
[1] | L. Block and J. Keesling, Computing the topological entropy of maps of the interval with three monotone pieces, J. Statist. Phys., 66 (1992), 755-774. doi: 10.1007/BF01055699. |
[2] | L. Block, J. Keesling, S. Li and K. Peterson, An improved algorithm for computing topological entropy, J. Statist. Phys., 55 (1989), 929-939. doi: 10.1007/BF01041072. |
[3] | H. Bruin and S. van Strien, Monotonicity of entropy for real multimodal maps, J. Amer. Math. Soc., 28 (2015), 1-61. doi: 10.1090/S0894-0347-2014-00795-5. |
[4] | S. P. Dawson, R. Galeeva, J. W. Milnor and C. Tresser, A monotonicity conjecture for real cubic maps, In Real and Complex Dynamical Systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., volume 464, Kluwer Acad. Publ., Dordrecht, 1995,165–183. |
[5] | A. Douady and J. H. Hubbard, Étude Dynamique des Polynômes Complexes. I & II, Publications Mathématiques d'Orsay, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984, 75 pp. |
[6] | A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297. doi: 10.1007/BF02392534. |
[7] | A. Douady, Topological entropy of unimodal maps: Monotonicity for quadratic polynomials, In Real and Complex Dynamical Systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., volume 464, Kluwer Acad. Publ., Dordrecht, 1995, 65–87. |
[8] | A. L. Epstein, Bounded hyperbolic components of quadratic rational maps, Ergodic Theory Dynam. Systems, 20 (2000), 727-748. doi: 10.1017/S0143385700000390. |
[9] | K. Filom, Real entropy rigidity under quasi-conformal deformations, preprint, 2018, arXiv: 1803.04082. |
[10] | K. Filom, Monotonicity of entropy for real quadratic rational maps, preprint, 2019, arXiv: 1901.03458. |
[11] | O. Kozlovski, On the structure of isentropes of real polynomials, J. Lond. Math. Soc. (2), 100 (2019), 159–182. doi: 10.1112/jlms.12207. |
[12] | G. Levin, W. Shen and S. van Strien, Positive transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps, preprint, 2019, arXiv: 1902.06732. |
[13] | R. Mañé, P. Sad and D. Sullivan On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4), 16 (1983), 193–217. doi: 10.24033/asens.1446. |
[14] | J. W. Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math., 2 (1993), 37-83. doi: 10.1080/10586458.1993.10504267. |
[15] | J. W. Milnor, On rational maps with two critical points, Experiment. Math., 9 (2000), 481-522. doi: 10.1080/10586458.2000.10504657. |
[16] | J. W. Milnor, Dynamics in One Complex Variable, 3$^{rd}$ edition, Annals of Mathematics Studies, volume 160, Princeton University Press, Princeton, NJ, 2006. |
[17] | J. W. Milnor, Hyperbolic components, In Conformal Dynamics and Hyperbolic Geometry, volume 573, Contemp. Math., Amer. Math. Soc., Providence, RI, 2012,183–232. doi: 10.1090/conm/573/11428. |
[18] | J. W. Milnor and W. P. Thurston, On iterated maps of the interval, In Dynamical Systems (College Park, MD, 1986–87), Lecture Notes in Math., volume 1342, Springer, Berlin, 1988,465–563. doi: 10.1007/BFb0082847. |
[19] | J. W. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys., 209 (2000), 123-178. doi: 10.1007/s002200050018. |
[20] | M. Misiurewicz, Continuity of entropy revisited, In Dynamical Systems and Applications, World Sci. Ser. Appl. Anal., volume 4, World Sci. Publ., River Edge, NJ, 1995,495–503. doi: 10.1142/9789812796417_0031. |
[21] | C. L. Petersen, On the Pommerenke-Levin-Yoccoz inequality, Ergodic Theory Dynam. Systems, 13 (1993), 785-806. |
[22] | K. M. Pilgrim, Cylinders for Iterated Rational Maps, Ph.D. Thesis, University of California, Berkeley, 1994,202 pp. |
[23] | K. M. Pilgrim, Rational maps whose Fatou components are Jordan domains, Ergodic Theory Dynam. Systems, 16 (1996), 1323-1343. doi: 10.1017/S0143385700010051. |
[24] | K. M. Pilgrim and L. Tan, Combining rational maps and controlling obstructions, Ergodic Theory Dynam. Systems, 18 (1998), 221-245. doi: 10.1017/S0143385798100329. |
[25] | M. Rees, Components of degree two hyperbolic rational maps, Invent. Math., 100 (1990), 357-382. doi: 10.1007/BF01231191. |
[26] | L. Tan, On pinching deformations of rational maps, Ann. Sci. École Norm. Sup. (4), 35 (2002), 353–370. doi: 10.1016/S0012-9593(02)01092-3. |
[27] | S. van Strien, Milnor's conjecture on monotonicity of topological entropy: Results and questions, In Frontiers in Complex Dynamics, Princeton Math. Ser., volume 51, Princeton Univ. Press, Princeton, NJ, 2014,323–337. |
The real moduli space
A colored version of Figure 1. The complement in
An entropy contour plot in the
An illustration of the real dynamics described in Proposition 3.2 in the case of
The dynamical
Dynamical plane of
An illustration of real bitransitive hyperbolic components adapted from [14,Figure 17]. Compare the limit points with Remark 3.10
The portion of the compactification
A graph of the real entropy along the post-critical line