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March  2022, 42(3): 1225-1241. doi: 10.3934/dcds.2021153

Statistics of multipliers for hyperbolic rational maps

Mathematics Institute, Zeeman Building, University of Warwick, CV4 7AL, Coventry, United Kingdom

* Corresponding author

Received  November 2020 Revised  August 2021 Published  March 2022 Early access  October 2021

In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.

Citation: Richard Sharp, Anastasios Stylianou. Statistics of multipliers for hyperbolic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1225-1241. doi: 10.3934/dcds.2021153
References:
[1]

A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, Springer-Verlag New York, 132, 1991. doi: 10.1007/978-1-4612-4422-6.

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 470, 2008. doi: 10.1007/978-3-540-77695-6.

[3]

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.

[4]

D. Dolgopyat, On decay of correlations in Anosov flows, Annals of Mathematics, Second Series, 147 (1998), 357-390.  doi: 10.2307/121012.

[5]

A. Eremenko and S. van Strien, Rational maps with real multipliers, Transactions of the American Mathematical Society, 363 (2011), 6453-6463.  doi: 10.1090/S0002-9947-2011-05308-0.

[6]

J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies, Third Edition 160, 2006, Princeton University Press. doi: 10.2307/j.ctt7rnxn.

[7]

F. Naud, Expanding maps on Cantor sets and analytic continuation of zeta functions, Annales Scientifiques de l'École Normale Supérieure, Série 4, 38 (2005), 116-153.  doi: 10.1016/j.ansens.2004.11.002.

[8]

H. Oh and D. Winter, Prime number theorems and holonomies for hyperbolic rational maps, Inventiones Mathematicae, 208 (2017), 401-440.  doi: 10.1007/s00222-016-0693-1.

[9]

W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque, 187–188, 1990.

[10]

Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, Journal of Statistical Physics, 86 (1997), 233-275.  doi: 10.1007/BF02180206.

[11]

M. Pollicott and R. Sharp, Rates of recurrence for $ \mathbb{Z}^q$ and $ \mathbb{R}^q$ extensions of subshifts of finite type, Journal of the London Mathematical Society, 49 (1994), 401-416.  doi: 10.1112/jlms/49.2.401.

[12]

M. Pollicott and R. Sharp, Correlations for pairs of closed geodesics, Inventiones Mathimaticae, 163 (2006), 1-24.  doi: 10.1007/s00222-004-0427-7.

[13]

M. Pollicott and R. Sharp, Distribution of ergodic sums for hyperbolic maps, Representation Theory, Dynamical Systems, and Asymptotic Combinatorics (ed. V. Kaimanovich and A. Lodkin), American Mathematical Society, (2006), 167–183. doi: 10.1090/trans2/217/11.

[14]

M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, American Journal of Mathematics, 120 (2008), 1019-1042.  doi: 10.1353/ajm.1998.0041.

[15]

M. Pollicott and R. Sharp, Correlations of length spectra for negatively curved manifolds, Communications in Mathematical Physics, 319 (2013), 515-533.  doi: 10.1007/s00220-012-1644-3.

[16]

D. Ruelle, The thermodynamic formalism for expanding maps, Communications in Mathematical Physics, 125 (1989), 239-262.  doi: 10.1007/BF01217908.

[17]

D. Ruelle, An extension of the theory of Fredholm determinants, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 175-193. 

[18]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511617546.

[19]

N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Berlin, New York: De Gruyter, 16, 1993. doi: 10.1515/9783110889314.

[20]

D. Sullivan, Conformal Dynamical Systems, Geometric Dynamics (ed. J. Palis), Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg, 1007 1983. doi: 10.1007/BFb0061443.

[21]

P. Wright, Ruelle's lemma and Ruelle zeta functions, Asymptotic Analysis, 80 (2012), 223-236.  doi: 10.3233/ASY-2012-1113.

[22]

M. Yu. Lyubich, Entropy of analytic endomorphisms of the Riemann sphere, Funktsional. Anal. i Prilozhen., 15 (1981), 83-84.  doi: 10.1017/S0143385700002030.

[23]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Inventiones Mathematicae, 99 (1990), 627-649.  doi: 10.1007/BF01234434.

show all references

References:
[1]

A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, Springer-Verlag New York, 132, 1991. doi: 10.1007/978-1-4612-4422-6.

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 470, 2008. doi: 10.1007/978-3-540-77695-6.

[3]

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.

[4]

D. Dolgopyat, On decay of correlations in Anosov flows, Annals of Mathematics, Second Series, 147 (1998), 357-390.  doi: 10.2307/121012.

[5]

A. Eremenko and S. van Strien, Rational maps with real multipliers, Transactions of the American Mathematical Society, 363 (2011), 6453-6463.  doi: 10.1090/S0002-9947-2011-05308-0.

[6]

J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies, Third Edition 160, 2006, Princeton University Press. doi: 10.2307/j.ctt7rnxn.

[7]

F. Naud, Expanding maps on Cantor sets and analytic continuation of zeta functions, Annales Scientifiques de l'École Normale Supérieure, Série 4, 38 (2005), 116-153.  doi: 10.1016/j.ansens.2004.11.002.

[8]

H. Oh and D. Winter, Prime number theorems and holonomies for hyperbolic rational maps, Inventiones Mathematicae, 208 (2017), 401-440.  doi: 10.1007/s00222-016-0693-1.

[9]

W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque, 187–188, 1990.

[10]

Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, Journal of Statistical Physics, 86 (1997), 233-275.  doi: 10.1007/BF02180206.

[11]

M. Pollicott and R. Sharp, Rates of recurrence for $ \mathbb{Z}^q$ and $ \mathbb{R}^q$ extensions of subshifts of finite type, Journal of the London Mathematical Society, 49 (1994), 401-416.  doi: 10.1112/jlms/49.2.401.

[12]

M. Pollicott and R. Sharp, Correlations for pairs of closed geodesics, Inventiones Mathimaticae, 163 (2006), 1-24.  doi: 10.1007/s00222-004-0427-7.

[13]

M. Pollicott and R. Sharp, Distribution of ergodic sums for hyperbolic maps, Representation Theory, Dynamical Systems, and Asymptotic Combinatorics (ed. V. Kaimanovich and A. Lodkin), American Mathematical Society, (2006), 167–183. doi: 10.1090/trans2/217/11.

[14]

M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, American Journal of Mathematics, 120 (2008), 1019-1042.  doi: 10.1353/ajm.1998.0041.

[15]

M. Pollicott and R. Sharp, Correlations of length spectra for negatively curved manifolds, Communications in Mathematical Physics, 319 (2013), 515-533.  doi: 10.1007/s00220-012-1644-3.

[16]

D. Ruelle, The thermodynamic formalism for expanding maps, Communications in Mathematical Physics, 125 (1989), 239-262.  doi: 10.1007/BF01217908.

[17]

D. Ruelle, An extension of the theory of Fredholm determinants, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 175-193. 

[18]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511617546.

[19]

N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Berlin, New York: De Gruyter, 16, 1993. doi: 10.1515/9783110889314.

[20]

D. Sullivan, Conformal Dynamical Systems, Geometric Dynamics (ed. J. Palis), Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg, 1007 1983. doi: 10.1007/BFb0061443.

[21]

P. Wright, Ruelle's lemma and Ruelle zeta functions, Asymptotic Analysis, 80 (2012), 223-236.  doi: 10.3233/ASY-2012-1113.

[22]

M. Yu. Lyubich, Entropy of analytic endomorphisms of the Riemann sphere, Funktsional. Anal. i Prilozhen., 15 (1981), 83-84.  doi: 10.1017/S0143385700002030.

[23]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Inventiones Mathematicae, 99 (1990), 627-649.  doi: 10.1007/BF01234434.

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