doi: 10.3934/dcds.2021153
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

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