The dependence of Lyapunov exponents of polynomials on their coefficients

Shrihari Sridharan; Atma Ram Tiwari

doi:10.3934/jcd.2019004

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2019, Volume 6, Issue 1: 95-109. Doi: 10.3934/jcd.2019004

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The dependence of Lyapunov exponents of polynomials on their coefficients

Shrihari Sridharan^, and
Atma Ram Tiwari

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

^* Corresponding author: Shrihari Sridharan
^* Corresponding author: Shrihari Sridharan

Published: August 2019

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B25, 37F10; Secondary: 37F15.

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