# American Institute of Mathematical Sciences

2021, 17: 465-479. doi: 10.3934/jmd.2021016

## Dynamics of transcendental Hénon maps III: Infinite entropy

 1 Dipartimento Di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 1, 00133 Roma, Italy 2 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze, 53/A, 43124 Parma, Italy 3 Department of Mathematical Sciences, NTNU Trondheim, O. S. Bragstads Plass 2E, 7034 Trondheim, Norway 4 Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands

Received  November 03, 2020 Revised  June 11, 2021 Published  November 2021

Fund Project: LA: Supported by the SIR grant "NEWHOLITE – New methods in holomorphic iteration" no. RBSI14CFME. Partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
AMB: This project has been partially supported by the project 'Transcendental Dynamics 1.5' inside the program FIL-Quota Incentivante of the University of Parma and co-sponsored by Fondazione Cariparma, and by Indam via the research group GNAMPA..

Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.

Citation: Leandro Arosio, Anna Miriam Benini, John Erik Fornæss, Han Peters. Dynamics of transcendental Hénon maps III: Infinite entropy. Journal of Modern Dynamics, 2021, 17: 465-479. doi: 10.3934/jmd.2021016
##### References:
 [1] L. Arosio, A. M. Benini, J. E. Fornæss and H. Peters, Dynamics of transcendental Hénon maps, Math. Ann., 373 (2019), 853-894.  doi: 10.1007/s00208-018-1643-6. [2] L. Arosio, A. M. Benini, J. E. Fornæss and H. Peters, Dynamics of transcendental Hénon maps II, preprint, arXiv: 1905.11557. [3] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9. [4] E. Bedford, M. Lyubich and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $\mathbf C^2$, Invent. Math. 114 (1993), no. 2,277–288., doi: 10.1007/BF01232671. [5] A. M. Benini, J. E. Fornæss and H. Peters, Entropy of transcendental entire functions, Ergodic Theory Dynam. Systems, 41 (2021), 338-348.  doi: 10.1017/etds.2019.65. [6] W. Bergweiler, The role of the Ahlfors five islands theorem in complex dynamics, Conform. Geom. Dyn., 4 (2000), 22-34.  doi: 10.1090/S1088-4173-00-00057-6. [7] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X. [8] R. Bowen, Erratum to "Entropy for group endomorphisms and homogeneous spaces", Trans. Amer. Math. Soc., 181 (1973), 509-510.  doi: 10.2307/1996650. [9] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X. [10] T.-C. Dinh and N. Sibony, Dynamique des applications d'allure polynomiale, J. Math. Pures Appl. 82 (2003), no. 4,367–423. doi: 10.1016/S0021-7824(03)00026-6. [11] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), 287–343. doi: 10.24033/asens.1491. [12] R. Dujardin, Hénon-like mappings in $\mathbb C^2$, Amer. J. Math., 126 (2004), 439-472.  doi: 10.1353/ajm.2004.0010. [13] M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), 217–235. [14] B. Hasselblatt, Z. Nitecki and J. Propp, Topological entropy for nonuniformly continuous maps, Discrete Contin. Dyn. Syst., 22 (2008), 201-213.  doi: 10.3934/dcds.2008.22.201. [15] J. E. Hofer, Topological entropy for noncompact spaces, Michigan Math. J., 21 (1974), 235-242.  doi: 10.1307/mmj/1029001311. [16] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-385.  doi: 10.1017/S0143385700002030. [17] M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. of Math. (2) 140 (1994), no. 2,347–404. doi: 10.2307/2118604. [18] M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2,425–457. doi: 10.1090/S0894-0347-1993-1182670-0. [19] J. L. Schiff, Normal Families, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0907-2. [20] J. Smillie, The entropy of polynomial diffeomorphisms of $\mathbb C^2$, Ergodic Theory Dynam. Systems, 10 (1990), 823-827.  doi: 10.1017/S0143385700005927. [21] M. Wendt, Invariante Maße und Juliamengen Meromorpher Funktionen, Diplomarbeit, University of Kiel, 2002. [22] M. Wendt, The entropy of entire transcendental functions, preprint, arXiv: 2011.02163. [23] M. Wendt, Zufällige Juliamengen und Invariante Maße mit Maximaler Entropie, Ph.D thesis, University of Kiel, https://macau.uni-kiel.de/receive/dissertation_diss_00001412 (German), 2005.

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##### References:
 [1] L. Arosio, A. M. Benini, J. E. Fornæss and H. Peters, Dynamics of transcendental Hénon maps, Math. Ann., 373 (2019), 853-894.  doi: 10.1007/s00208-018-1643-6. [2] L. Arosio, A. M. Benini, J. E. Fornæss and H. Peters, Dynamics of transcendental Hénon maps II, preprint, arXiv: 1905.11557. [3] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9. [4] E. Bedford, M. Lyubich and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $\mathbf C^2$, Invent. Math. 114 (1993), no. 2,277–288., doi: 10.1007/BF01232671. [5] A. M. Benini, J. E. Fornæss and H. Peters, Entropy of transcendental entire functions, Ergodic Theory Dynam. Systems, 41 (2021), 338-348.  doi: 10.1017/etds.2019.65. [6] W. Bergweiler, The role of the Ahlfors five islands theorem in complex dynamics, Conform. Geom. Dyn., 4 (2000), 22-34.  doi: 10.1090/S1088-4173-00-00057-6. [7] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X. [8] R. Bowen, Erratum to "Entropy for group endomorphisms and homogeneous spaces", Trans. Amer. Math. Soc., 181 (1973), 509-510.  doi: 10.2307/1996650. [9] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X. [10] T.-C. Dinh and N. Sibony, Dynamique des applications d'allure polynomiale, J. Math. Pures Appl. 82 (2003), no. 4,367–423. doi: 10.1016/S0021-7824(03)00026-6. [11] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), 287–343. doi: 10.24033/asens.1491. [12] R. Dujardin, Hénon-like mappings in $\mathbb C^2$, Amer. J. Math., 126 (2004), 439-472.  doi: 10.1353/ajm.2004.0010. [13] M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), 217–235. [14] B. Hasselblatt, Z. Nitecki and J. Propp, Topological entropy for nonuniformly continuous maps, Discrete Contin. Dyn. Syst., 22 (2008), 201-213.  doi: 10.3934/dcds.2008.22.201. [15] J. E. Hofer, Topological entropy for noncompact spaces, Michigan Math. J., 21 (1974), 235-242.  doi: 10.1307/mmj/1029001311. [16] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-385.  doi: 10.1017/S0143385700002030. [17] M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. of Math. (2) 140 (1994), no. 2,347–404. doi: 10.2307/2118604. [18] M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2,425–457. doi: 10.1090/S0894-0347-1993-1182670-0. [19] J. L. Schiff, Normal Families, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0907-2. [20] J. Smillie, The entropy of polynomial diffeomorphisms of $\mathbb C^2$, Ergodic Theory Dynam. Systems, 10 (1990), 823-827.  doi: 10.1017/S0143385700005927. [21] M. Wendt, Invariante Maße und Juliamengen Meromorpher Funktionen, Diplomarbeit, University of Kiel, 2002. [22] M. Wendt, The entropy of entire transcendental functions, preprint, arXiv: 2011.02163. [23] M. Wendt, Zufällige Juliamengen und Invariante Maße mit Maximaler Entropie, Ph.D thesis, University of Kiel, https://macau.uni-kiel.de/receive/dissertation_diss_00001412 (German), 2005.
Illustration of the statement and proof of Lemma 3.12. The disks $\mathbb D_r(x_i)$ are contained in larger disks $\mathbb D_R(x_i)$, which do not appear in this picture
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