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

[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

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

[1]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[2]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[3]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[4]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[5]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[6]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[7]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[8]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[9]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[10]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[11]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117

[12]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[13]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[14]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[15]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

[16]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[17]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[18]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[19]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[20]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

 Impact Factor: 

Metrics

  • PDF downloads (114)
  • HTML views (547)
  • Cited by (0)

Other articles
by authors

[Back to Top]