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]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[2]

Sergey Zelik. On the Lyapunov dimension of cascade systems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 971-985. doi: 10.3934/cpaa.2008.7.971

[3]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187

[4]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[5]

Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807

[6]

Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166

[7]

Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224

[8]

Xueting Tian, Shirou Wang, Xiaodong Wang. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1019-1032. doi: 10.3934/dcds.2019042

[9]

P. Kaplický, Dalibor Pražák. Lyapunov exponents and the dimension of the attractor for 2d shear-thinning incompressible flow. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 961-974. doi: 10.3934/dcds.2008.20.961

[10]

Artur Avila. Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension. Journal of Modern Dynamics, 2009, 3 (4) : 631-636. doi: 10.3934/jmd.2009.3.631

[11]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91

[12]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957

[13]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107

[14]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433

[15]

Jifeng Chu, Jinzhi Lei, Meirong Zhang. Lyapunov stability for conservative systems with lower degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 423-443. doi: 10.3934/dcdsb.2011.16.423

[16]

Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303

[17]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060

[18]

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Kinematical structural stability. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 529-536. doi: 10.3934/dcdss.2016010

[19]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287

[20]

Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487

 Impact Factor: 

Metrics

  • PDF downloads (86)
  • HTML views (286)
  • Cited by (0)

Other articles
by authors

[Back to Top]