# American Institute of Mathematical Sciences

2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353

## Typical dynamics of plane rational maps with equal degrees

 1 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States, United States 2 IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202, United States

Received  January 2016 Revised  April 2016 Published  August 2016

Let $f:\mathbb{CP}^2⇢\mathbb{CP}^2$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle type and one of repelling type. Neither measure is supported in an algebraic curve, and $f_T$ is 'fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation of $\mathbb{C}\mathbb{P}^2$. In fact, absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets of $Aut(\mathbb{P}^2)$. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.
Citation: Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. Journal of Modern Dynamics, 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353
##### References:
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Projective and inductive limits of polynomials, in Real and Complex Dynamical Systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995, 89-132.  Google Scholar [28] M. Jonsson, Hyperbolic dynamics of endomorphisms,, Unpublished note: , ().   Google Scholar [29] M. Jonsson and B. Weickert, A nonalgebraic attractor in $\mathbbP^2$, Proc. Amer. Math. Soc., 128 (2000), 2999-3002. doi: 10.1090/S0002-9939-00-05529-5.  Google Scholar [30] S. Kashner, R. Pérez and R. Roeder, Examples of rational maps of $\mathbb{CP}^2$ with equal dynamical degrees and no invariant foliation, Bulletin de la SMF, 144 (2016), 279-297. Google Scholar [31] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar [32] B. P. 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##### References:
 [1] L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova, 231 (2000), 5-45.  Google Scholar [2] L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets, in Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 25-118.  Google Scholar [3] E. Bedford, S. Cantat and K. Kim, Pseudo-automorphisms with no invariant foliation, J. Mod. Dyn., 8 (2014), 221-250. doi: 10.3934/jmd.2014.8.221.  Google Scholar [4] E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of $C^2$. IV. The measure of maximal entropy and laminar currents, Invent. Math., 112 (1993), 77-125. doi: 10.1007/BF01232426.  Google Scholar [5] J.-Y. Briend and J. Duval, Deux caractérisations de la mesure d'équilibre d'un endomorphisme de $P^k(\mathbbC)$, Publ. Math. Inst. Hautes Études Sci., (2001), 145-159. doi: 10.1007/s10240-001-8190-4.  Google Scholar [6] S. Cantat, Dynamique des automorphismes des surfaces $K3$, Acta Math., 187 (2001), 1-57. doi: 10.1007/BF02392831.  Google Scholar [7] S. Daurat, On the size of attractors in $\mathbbP^k$, Math. Z., 277 (2014), 629-650. doi: 10.1007/s00209-013-1269-z.  Google Scholar [8] H. De Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.), (2010), vi+98.  Google Scholar [9] J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., 123 (2001), 1135-1169. doi: 10.1353/ajm.2001.0038.  Google Scholar [10] J. Diller, R. Dujardin and V. Guedj, Dynamics of meromorphic maps with small topological degree I: From cohomology to currents, Indiana Univ. Math. J., 59 (2010), 521-561. doi: 10.1512/iumj.2010.59.4023.  Google Scholar [11] J. Diller, R. Dujardin and V. Guedj, Dynamics of meromorphic maps with small topological degree III: Geometric currents and ergodic theory Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 235-278.  Google Scholar [12] J. Diller, R. Dujardin and V. Guedj, Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure, Comment. Math. Helv., 86 (2011), 277-316. doi: 10.4171/CMH/224.  Google Scholar [13] J. Diller and J.-L. Lin, Rational surface maps with invariant meromorphic two-forms, Math. Ann., 364 (2016), 313-352. doi: 10.1007/s00208-015-1211-2.  Google Scholar [14] T.-C. Dinh, Attracting current and equilibrium measure for attractors on $\mathbbP^k$, J. Geom. Anal., 17 (2007), 227-244. doi: 10.1007/BF02930722.  Google Scholar [15] T.-C. Dinh, V.-A. Nguyên and T. Truong, Equidistribution for meromorphic maps with dominant topological degree, Indiana Univ. Math. J., 64 (2015), 1805-1828.  Google Scholar [16] T.-C. Dinh and N. Sibony, Une borne supêrieure pour l'entropie topologique d'une application rationnelle, Ann. of Math. (2), 161 (2005), 1637-1644. doi: 10.4007/annals.2005.161.1637.  Google Scholar [17] R. Dujardin, Hênon-like mappings in $\mathbbC^2$, Amer. J. Math., 126 (2004), 439-472.  Google Scholar [18] C. Favre and J. V. Pereira, Foliations invariant by rational maps, Math. Z., 268 (2011), 753-770. doi: 10.1007/s00209-010-0693-6.  Google Scholar [19] J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension. II, in Modern Methods in Complex Analysis (Princeton, NJ, 1992), Ann. of Math. Stud., 137, Princeton Univ. Press, Princeton, NJ, 1995, 135-182.  Google Scholar [20] J. E. Fornaess and N. Sibony, Dynamics of $P^2$ (examples), in Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998), Contemp. Math., 269, Amer. Math. Soc., Providence, RI, 2001, 47-85. doi: 10.1090/conm/269/04329.  Google Scholar [21] J. E. Fornaess and B. Weickert, Attractors in $P^2$, in Several Complex Variables (Berkeley, CA, 1995-1996), Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999, 297-307.  Google Scholar [22] R. I. Grigorchuk and A. Żuk, The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geom. Dedicata, 87 (2001), 209-244. doi: 10.1023/A:1012061801279.  Google Scholar [23] M. Gromov, Entropy, homology and semialgebraic geometry, Séminaire Bourbaki, Vol. 1985/86, Astérisque, No. 145-146 (1987), 5, 225-240.  Google Scholar [24] V. Guedj, Entropie topologique des applications méromorphes, Ergodic Theory Dynam. Systems, 25 (2005), 1847-1855. doi: 10.1017/S0143385705000192.  Google Scholar [25] V. Guedj, Ergodic properties of rational mappings with large topological degree, Ann. of Math. (2), 161 (2005), 1589-1607. doi: 10.4007/annals.2005.161.1589.  Google Scholar [26] V. Guedj, Propriétés ergodiques des applications rationnelles, in Quelques aspects des systèmes dynamiques polynomiaux, Panor. Synthèses, 30, Soc. Math. France, Paris, 2010, 97-202.  Google Scholar [27] J. H. Hubbard and R. W. Oberste-Vorth, Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, in Real and Complex Dynamical Systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995, 89-132.  Google Scholar [28] M. Jonsson, Hyperbolic dynamics of endomorphisms,, Unpublished note: , ().   Google Scholar [29] M. Jonsson and B. Weickert, A nonalgebraic attractor in $\mathbbP^2$, Proc. Amer. Math. Soc., 128 (2000), 2999-3002. doi: 10.1090/S0002-9939-00-05529-5.  Google Scholar [30] S. Kashner, R. Pérez and R. Roeder, Examples of rational maps of $\mathbb{CP}^2$ with equal dynamical degrees and no invariant foliation, Bulletin de la SMF, 144 (2016), 279-297. Google Scholar [31] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar [32] B. P. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts, Universitext, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58822-8.  Google Scholar [33] S. G. Krantz, Complex Analysis: The Geometric Viewpoint, Carus Mathematical Monographs, 23, Mathematical Association of America, Washington, DC, 1990.  Google Scholar [34] S. Lang, Introduction to Complex Hyperbolic Spaces, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1945-1.  Google Scholar [35] H. Liu, A Plane Rational Map with Chebyshev-like Dynamics, Ph.D. thesis, University of Notre Dame, 2014.  Google Scholar [36] W. Parry, Entropy and Generators in Ergodic Theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969.  Google Scholar [37] F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139193184.  Google Scholar [38] C. Sabot, Spectral properties of self-similar lattices and iteration of rational maps, Mém. Soc. Math. Fr. (N.S.), (2003), vi+104.  Google Scholar [39] G. Vigny, Hyperbolic measure of maximal entropy for generic rational maps of $\mathbbP^k$, Ann. Inst. Fourier (Grenoble), 64 (2014), 645-680. doi: 10.5802/aif.2861.  Google Scholar
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