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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

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