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November  2022, 42(11): 5377-5386. doi: 10.3934/dcds.2022101

Monotonicity of entropy for unimodal real quadratic rational maps

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China

*Corresponding author: Yan Gao

Received  September 2021 Revised  April 2022 Published  November 2022 Early access  August 2022

Fund Project: The author is supported by NSFC grants 11871354 and 12131016

We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients and real critical points. This confirms a conjecture made in [8].

Citation: Yan Gao. Monotonicity of entropy for unimodal real quadratic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (11) : 5377-5386. doi: 10.3934/dcds.2022101
References:
[1]

L. Bers and H. L. Royden, Holomorphic families of injections, Acta Math., 157 (1986), 259-286.  doi: 10.1007/BF02392595.

[2]

A. BonifantJ. Milnor and S. Sutherland, The W. Thurston algorithm applied to real polynomial maps, Conform. Geom. Dyn., 25 (2021), 179-199.  doi: 10.1090/ecgd/365.

[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. Milnor and C. Tresser, A monotonicity conjecture for real cubic maps, In Real and Complex Dynamical Systems, 1993 (Hillerød), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995,165–183.

[5]

W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 25, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.

[6]

A. Douady, Topological entropy of unimodal maps: Monotonicity for quadratic polynomials, In Real and Complex Dynamical Systems, 1993 (Hillerød), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995, 65–87.

[7]

A. Douady and J. H. Hubbard, Étude Dynamique des Polynômes Complexes, Partie I, Publications Mathématiques d'Orsay, 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.

[8]

K. Filom, Monotonicity of entropy for real quadratic rational maps, Nonlinearity, 34 (2021), 6587-6626.  doi: 10.1088/1361-6544/ac15aa.

[9]

K. Filom, Real entropy rigidity under quasi-conformal deformations, Conform. Geom. Dyn., 25 (2021), 1-33.  doi: 10.1090/ecgd/356.

[10]

K. Filom and K. M. Pilgrim, On the non-monotonicity of entropy for a class of real quadratic rational maps, Journal of Modern Dynamics, 16 (2020), 225-254.  doi: 10.3934/jmd.2020008.

[11]

O. Kozlovski, On the structure of isentropes of real polynomials, J. Lond. Math. Soc., 100 (2019), 159-182.  doi: 10.1112/jlms.12207.

[12]

G. Levin, W. Shen and S. van Strien, Transversality for critical relations of families of rational maps: An elementary proof, In New Trends in One-Dimensional Dynamics, Springer Proc. Math. Stat., 285, Springer, Cham., 2019,201–220. doi: 10.1007/978-3-030-16833-9_11.

[13]

G. LevinW. Shen and S. van Strien, Positive transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps, Nonlinearity, 33 (2020), 3970-4012.  doi: 10.1088/1361-6544/ab853e.

[14]

J. Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math., 2 (1993), 37-83.  doi: 10.1080/10586458.1993.10504267.

[15]

J. Milnor and W. Thurston, On iterated maps of the interval, In Dynamical Systems, 1986–87 (College Park, MD), Lecture Notes in Math., 1342, Springer, Berlin, 1988,465–563. doi: 10.1007/BFb0082847.

[16]

J. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys., 209 (2000), 123-178.  doi: 10.1007/s002200050018.

[17]

S. van Strien, Milnor's conjecture on monotonicity of topological entropy: Results and questions, Frontiers in Complex Dynamics, Princeton Math. Ser., 51, Princeton Univ. Press, Princeton, NJ, 2014,323-337.

show all references

References:
[1]

L. Bers and H. L. Royden, Holomorphic families of injections, Acta Math., 157 (1986), 259-286.  doi: 10.1007/BF02392595.

[2]

A. BonifantJ. Milnor and S. Sutherland, The W. Thurston algorithm applied to real polynomial maps, Conform. Geom. Dyn., 25 (2021), 179-199.  doi: 10.1090/ecgd/365.

[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. Milnor and C. Tresser, A monotonicity conjecture for real cubic maps, In Real and Complex Dynamical Systems, 1993 (Hillerød), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995,165–183.

[5]

W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 25, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.

[6]

A. Douady, Topological entropy of unimodal maps: Monotonicity for quadratic polynomials, In Real and Complex Dynamical Systems, 1993 (Hillerød), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995, 65–87.

[7]

A. Douady and J. H. Hubbard, Étude Dynamique des Polynômes Complexes, Partie I, Publications Mathématiques d'Orsay, 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.

[8]

K. Filom, Monotonicity of entropy for real quadratic rational maps, Nonlinearity, 34 (2021), 6587-6626.  doi: 10.1088/1361-6544/ac15aa.

[9]

K. Filom, Real entropy rigidity under quasi-conformal deformations, Conform. Geom. Dyn., 25 (2021), 1-33.  doi: 10.1090/ecgd/356.

[10]

K. Filom and K. M. Pilgrim, On the non-monotonicity of entropy for a class of real quadratic rational maps, Journal of Modern Dynamics, 16 (2020), 225-254.  doi: 10.3934/jmd.2020008.

[11]

O. Kozlovski, On the structure of isentropes of real polynomials, J. Lond. Math. Soc., 100 (2019), 159-182.  doi: 10.1112/jlms.12207.

[12]

G. Levin, W. Shen and S. van Strien, Transversality for critical relations of families of rational maps: An elementary proof, In New Trends in One-Dimensional Dynamics, Springer Proc. Math. Stat., 285, Springer, Cham., 2019,201–220. doi: 10.1007/978-3-030-16833-9_11.

[13]

G. LevinW. Shen and S. van Strien, Positive transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps, Nonlinearity, 33 (2020), 3970-4012.  doi: 10.1088/1361-6544/ab853e.

[14]

J. Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math., 2 (1993), 37-83.  doi: 10.1080/10586458.1993.10504267.

[15]

J. Milnor and W. Thurston, On iterated maps of the interval, In Dynamical Systems, 1986–87 (College Park, MD), Lecture Notes in Math., 1342, Springer, Berlin, 1988,465–563. doi: 10.1007/BFb0082847.

[16]

J. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys., 209 (2000), 123-178.  doi: 10.1007/s002200050018.

[17]

S. van Strien, Milnor's conjecture on monotonicity of topological entropy: Results and questions, Frontiers in Complex Dynamics, Princeton Math. Ser., 51, Princeton Univ. Press, Princeton, NJ, 2014,323-337.

Figure 1.  An entropy contour plot for the unimodal quadratic rational maps 8 adapted from [8]. The ordering of colors is black$ < $blue$ < $magenta$ < $green$ < $cyan$ < $yellow$ < $red and they correspond to the partition $ [0, 0.1) $, $ [0.1, 0.25) $, $ [0.25, 0.4) $, $ [0.4, 0.48) $, $ [0.48, 0.55) $, $ [0.55, 0.65) $ and $ [0.65, \log(2)] $ of $ [0, \log(2)\approx 0.7] $. The connectedness of entropy level sets here suggests the monotonicity of entropy in the unimodal region of $ {\rm rat}_2({\mathbb R}) $
Figure 2.  A schematic picture of bones (in black) defined by the post-critical relations $ f^{ n}(c_1) = c_1 $ in the unimodal region of the $ (\sigma_1, \sigma_2) $-plane. They are disjoint one-dimensional submanifolds. There are bone-arcs connecting PCF quadratic polynomials to PCF quadratic rational maps on the $ f(c_1) = c_2 $ line. Any other bone in this region should be a Jordan curve, a bone-loop. The picture is adapted from [8]
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