June  2019, 6(1): 95-109. doi: 10.3934/jcd.2019004

The dependence of Lyapunov exponents of polynomials on their coefficients

Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM), Maruthamala P.O., Vithura, Thiruvananthapuram, India. PIN 695 551

* Corresponding author: Shrihari Sridharan

Published  July 2019

Fund Project: The first author was supported by a Fasttrack Grant for Young Scientists awarded by the Department of Science and Technology, Government of India, vide SR/FTP/MS-008/2012.

In this paper, we consider the family of hyperbolic quadratic polynomials parametrised by a complex constant; namely $ P_{c} (z) = z^{2} + c $ with $ |c| < 1 $ and the family of hyperbolic cubic polynomials parametrised by two complex constants; namely $ P_{(a_{1}, \, a_{0})} (z) = z^{3} + a_{1} z + a_{0} $ with $ |a_{i}| < 1 $, restricted on their respective Julia sets. We compute the Lyapunov characteristic exponents for these polynomial maps over corresponding Julia sets, with respect to various Bernoulli measures and obtain results pertaining to the dependence of the behaviour of these exponents on the parameters describing the polynomial map. We achieve this using the theory of thermodynamic formalism, the pressure function in particular.

Citation: Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
References:
[1]

A. F. Beardon, Iteration of Rational Functions, Complex Analytic Dynamical Systems, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991.  Google Scholar

[2]

G. BenettinL. Galgani and J.-M. Strelcyn, Kolmogorov entropy and numerical experiments, Physical Review A, 14 (1976), 2338-2345.  doi: 10.1103/PhysRevA.14.2338.  Google Scholar

[3]

A. BlokhL. OversteegenR. Ptacek and V. Timorin, Quadratic-like dynamics of cubic polynomials, Comm. Math. Phys., 341 (2016), 733-749.  doi: 10.1007/s00220-015-2559-6.  Google Scholar

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A. BlokhL. OversteegenR. Ptacek and V. Timorin, The main cubioid, Nonlinearity, 27 (2014), 1879-1897.  doi: 10.1088/0951-7715/27/8/1879.  Google Scholar

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L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[6]

Z. Coelho and W. Parry, Central limit asymptotics for shifts of finite type, Israel J. Math., 69 (1990), 235-249.  doi: 10.1007/BF02937307.  Google Scholar

[7]

M. Denker and M. Urbanski, Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4 (1991), 103-134.  doi: 10.1088/0951-7715/4/1/008.  Google Scholar

[8]

A. GarijoX. Jarque and J. Villadelprat, An effective algorithm to compute Mandelbrot sets in parameter planes, Numer. Algorithms, 76 (2017), 555-571.  doi: 10.1007/s11075-017-0270-8.  Google Scholar

[9]

M. Yu. Lyubich, The dynamics of rational transforms: The topological picture, (Russian) Uspekhi Mat. Nauk., 41 (1986), 35–95; Russian Math. Surveys, 41 (1986), 43–117. doi: 10.1070/RM1986v041n04ABEH003376.  Google Scholar

[10]

A. Manning, The dimension of the maximal measure for a polynomial map, Ann. of Math., 119 (1984), 425-430.  doi: 10.2307/2007044.  Google Scholar

[11]

Ya. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, (Russian) Uspekhi Mat. Nauk., 32 (1977), 55–112; Russian Math. Surveys, 32 (1977), 55–114. doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

[12]

D. Ruelle, Thermodynamic Formalism, Encyclopedia Mathematics and its Applications, Reading: Addison-Wesley, 1978.  Google Scholar

[13]

S. Sridharan, Non-vanishing derivatives of Lyapunov exponents and the pressure function, Dyn. Syst., 21 (2006), 491-500.  doi: 10.1080/14689360600872037.  Google Scholar

[14]

N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, De Gruyter studies in Mathematics, 16, Walter de Gruyter and Co., Berlin, 1993.  Google Scholar

[15]

P. Walters, An Introduction to Ergodic Theory, Graduate texts in Mathematics, 79, Springer-Verlag, New York, 1982.  Google Scholar

[16]

M. Zinsmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems, Panoramas and Synthesis, (French) [Formalisme thermodynamique et systémes dynamiques holomorphes], Panoramas et Synthéses, Société Mathématique de France, 4, 1996.  Google Scholar

show all references

References:
[1]

A. F. Beardon, Iteration of Rational Functions, Complex Analytic Dynamical Systems, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991.  Google Scholar

[2]

G. BenettinL. Galgani and J.-M. Strelcyn, Kolmogorov entropy and numerical experiments, Physical Review A, 14 (1976), 2338-2345.  doi: 10.1103/PhysRevA.14.2338.  Google Scholar

[3]

A. BlokhL. OversteegenR. Ptacek and V. Timorin, Quadratic-like dynamics of cubic polynomials, Comm. Math. Phys., 341 (2016), 733-749.  doi: 10.1007/s00220-015-2559-6.  Google Scholar

[4]

A. BlokhL. OversteegenR. Ptacek and V. Timorin, The main cubioid, Nonlinearity, 27 (2014), 1879-1897.  doi: 10.1088/0951-7715/27/8/1879.  Google Scholar

[5]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[6]

Z. Coelho and W. Parry, Central limit asymptotics for shifts of finite type, Israel J. Math., 69 (1990), 235-249.  doi: 10.1007/BF02937307.  Google Scholar

[7]

M. Denker and M. Urbanski, Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4 (1991), 103-134.  doi: 10.1088/0951-7715/4/1/008.  Google Scholar

[8]

A. GarijoX. Jarque and J. Villadelprat, An effective algorithm to compute Mandelbrot sets in parameter planes, Numer. Algorithms, 76 (2017), 555-571.  doi: 10.1007/s11075-017-0270-8.  Google Scholar

[9]

M. Yu. Lyubich, The dynamics of rational transforms: The topological picture, (Russian) Uspekhi Mat. Nauk., 41 (1986), 35–95; Russian Math. Surveys, 41 (1986), 43–117. doi: 10.1070/RM1986v041n04ABEH003376.  Google Scholar

[10]

A. Manning, The dimension of the maximal measure for a polynomial map, Ann. of Math., 119 (1984), 425-430.  doi: 10.2307/2007044.  Google Scholar

[11]

Ya. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, (Russian) Uspekhi Mat. Nauk., 32 (1977), 55–112; Russian Math. Surveys, 32 (1977), 55–114. doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

[12]

D. Ruelle, Thermodynamic Formalism, Encyclopedia Mathematics and its Applications, Reading: Addison-Wesley, 1978.  Google Scholar

[13]

S. Sridharan, Non-vanishing derivatives of Lyapunov exponents and the pressure function, Dyn. Syst., 21 (2006), 491-500.  doi: 10.1080/14689360600872037.  Google Scholar

[14]

N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, De Gruyter studies in Mathematics, 16, Walter de Gruyter and Co., Berlin, 1993.  Google Scholar

[15]

P. Walters, An Introduction to Ergodic Theory, Graduate texts in Mathematics, 79, Springer-Verlag, New York, 1982.  Google Scholar

[16]

M. Zinsmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems, Panoramas and Synthesis, (French) [Formalisme thermodynamique et systémes dynamiques holomorphes], Panoramas et Synthéses, Société Mathématique de France, 4, 1996.  Google Scholar

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