Abstract
We show that describing rational functions \(f_1,\) \(f_2,\) \(\dots ,f_n\) sharing the measure of maximal entropy reduces to describing solutions of the functional equation \(A\circ X_1=A\circ X_2=\dots =A\circ X_n\) in rational functions. We also provide some results about solutions of this equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An, T.T.H., Diep, N.T.N.: Genus one factors of curves defined by separated variable polynomials. J. Number Theory 133(8), 2616–2634 (2013)
Atela, P.: Sharing a Julia set: the polynomial case, Progress in Holomorphic Dynamics, pp. 102–115. Pitman Res. Notes Math. Ser., vol. 387. Longman, Harlow (1998)
Atela, P., Hu, J.: Commuting polynomials and polynomials with same Julia set. Int. J. Bifurc. Chaos Appl. Sci. Eng. 6(12A), 2427–2432 (1996)
Avanzi, R., Zannier, U.: Genus one curves defined by separated variable polynomials and a polynomial Pell equation. Acta Arith. 99, 227–256 (2001)
Avanzi, R., Zannier, U.: The equation \(f(X)=f(Y)\) in rational functions \(X=X(t), Y=Y(t),\). Composit. Math. 139(3), 263–295 (2003)
Baker, I., Eremenko, A.: A problem on Julia sets. Ann. Acad. Sci. Fennicae (series A.I. Math.) 12, 229–236 (1987)
Beardon, A.: Symmetries of Julia sets. Bull. Lond. Math. Soc. 22(6), 576–582 (1990)
Beardon, A.: Polynomials with identical Julia sets. Complex Var. Theory Appl. 17(3–4), 195–200 (1992)
Beardon, A., Ng, T.W.: Parametrizations of algebraic curves. Ann. Acad. Sci. Fenn. Math. 31(2), 541–554 (2006)
Bilu, Y., Tichy, R.: The Diophantine equation \(f(x) = g(y)\). Acta Arith. 95(3), 261–288 (2000)
Freire, A., Lopes, A., Mañé, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14(1), 45–62 (1983)
Fried, M.: On a theorem of Ritt and related diophantine problems. J. Reine Angew. Math. 264, 40–55 (1973)
Levin, G.: Symmetries on Julia sets. Math. Notes 48(5–6), 1126–1131 (1990)
Levin, G., Przytycki, F.: When do two rational functions have the same Julia set? Proc. Am. Math. Soc. 125(7), 2179–2190 (1997)
Ljubich, M.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3(3), 351–385 (1983)
Lyubich, M., Minsky, Y.: Laminations in holomorphic dynamics. J. Differ. Geom. 47(1), 17–94 (1997)
Ng, T.W., Wang, M.X.: Ritt’s theory on the unit disk. Forum Math. 25(4), 821–851 (2013)
Pakovich, F.: On polynomials sharing preimages of compact sets, and related questions. Geom. Funct. Anal. 18(1), 163–183 (2008)
Pakovich, F.: Prime and composite Laurent polynomials. Bull. Sci. Math. 133, 693–732 (2009)
Pakovich, F.: On the equation P(f)=Q(g), where P, Q are polynomials and f, g are entire functions. Am. J. Math. 132(6), 1591–1607 (2010)
Pakovich, F.: On rational functions whose normalization has genus zero or one. Acta. Arith. 182(1), 73–100 (2018a)
Pakovich, F.: On algebraic curves A(x)-B(y)=0 of genus zero. Math. Z. 288(1), 299–310 (2018b)
Pakovich, F.: Semiconjugate rational functions: a dynamical approach. Arnold Math. J. 4(1), 59–68 (2018c)
Pakovich, F.: Recomposing rational functions. Int. Math. Res. Not. 7, 1921–1935 (2019a)
Pakovich, F.: Commuting rational functions revisited. Ergod. Theory Dyn. Syst. (2019b). https://doi.org/10.1017/etds.2019.51
Ritt, J.: Prime and composite polynomials. Am. M. S. Trans. 23, 51–66 (1922)
Ritt, J.: Permutable rational functions. Trans. Am. Math. Soc. 25, 399–448 (1923)
Ritt, J.: Equivalent rational substitutions. Trans. Am. Math. Soc. 26(2), 221–229 (1924)
Schmidt, W., Steinmetz, N.: The polynomials associated with a Julia set. Bull. Lond. Math. Soc. 27(3), 239–241 (1995)
Ye, H.: Rational functions with identical measure of maximal entropy. Adv. Math. 268, 373–395 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
To Misha Lyubich, on the occasion of his 60th birthday.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was supported by ISF Grant No. 1432/18.
Rights and permissions
About this article
Cite this article
Pakovich, F. On Rational Functions Sharing the Measure of Maximal Entropy. Arnold Math J. 6, 387–396 (2020). https://doi.org/10.1007/s40598-020-00141-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40598-020-00141-z