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:
[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.

[2]

L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets,, in Fractals in Graz 2001, (2001), 25.

[3]

E. Bedford, S. Cantat and K. Kim, Pseudo-automorphisms with no invariant foliation,, J. Mod. Dyn., 8 (2014), 221. doi: 10.3934/jmd.2014.8.221.

[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. doi: 10.1007/BF01232426.

[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. doi: 10.1007/s10240-001-8190-4.

[6]

S. Cantat, Dynamique des automorphismes des surfaces $K3$,, Acta Math., 187 (2001), 1. doi: 10.1007/BF02392831.

[7]

S. Daurat, On the size of attractors in $\mathbbP^k$,, Math. Z., 277 (2014), 629. doi: 10.1007/s00209-013-1269-z.

[8]

H. De Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps,, Mém. Soc. Math. Fr. (N.S.), (2010).

[9]

J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces,, Amer. J. Math., 123 (2001), 1135. doi: 10.1353/ajm.2001.0038.

[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. doi: 10.1512/iumj.2010.59.4023.

[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.

[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. doi: 10.4171/CMH/224.

[13]

J. Diller and J.-L. Lin, Rational surface maps with invariant meromorphic two-forms,, Math. Ann., 364 (2016), 313. doi: 10.1007/s00208-015-1211-2.

[14]

T.-C. Dinh, Attracting current and equilibrium measure for attractors on $\mathbbP^k$,, J. Geom. Anal., 17 (2007), 227. doi: 10.1007/BF02930722.

[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.

[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. doi: 10.4007/annals.2005.161.1637.

[17]

R. Dujardin, Hênon-like mappings in $\mathbbC^2$,, Amer. J. Math., 126 (2004), 439.

[18]

C. Favre and J. V. Pereira, Foliations invariant by rational maps,, Math. Z., 268 (2011), 753. doi: 10.1007/s00209-010-0693-6.

[19]

J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension. II,, in Modern Methods in Complex Analysis (Princeton, (1992), 135.

[20]

J. E. Fornaess and N. Sibony, Dynamics of $P^2$ (examples),, in Laminations and Foliations in Dynamics, (1998), 47. doi: 10.1090/conm/269/04329.

[21]

J. E. Fornaess and B. Weickert, Attractors in $P^2$,, in Several Complex Variables (Berkeley, (1999), 1995.

[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. doi: 10.1023/A:1012061801279.

[23]

M. Gromov, Entropy, homology and semialgebraic geometry,, Séminaire Bourbaki, (1987), 145.

[24]

V. Guedj, Entropie topologique des applications méromorphes,, Ergodic Theory Dynam. Systems, 25 (2005), 1847. doi: 10.1017/S0143385705000192.

[25]

V. Guedj, Ergodic properties of rational mappings with large topological degree,, Ann. of Math. (2), 161 (2005), 1589. doi: 10.4007/annals.2005.161.1589.

[26]

V. Guedj, Propriétés ergodiques des applications rationnelles,, in Quelques aspects des systèmes dynamiques polynomiaux, (2010), 97.

[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), 89.

[28]

M. Jonsson, Hyperbolic dynamics of endomorphisms,, Unpublished note: , ().

[29]

M. Jonsson and B. Weickert, A nonalgebraic attractor in $\mathbbP^2$,, Proc. Amer. Math. Soc., 128 (2000), 2999. doi: 10.1090/S0002-9939-00-05529-5.

[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.

[31]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187.

[32]

B. P. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts,, Universitext, (1998). doi: 10.1007/978-3-642-58822-8.

[33]

S. G. Krantz, Complex Analysis: The Geometric Viewpoint,, Carus Mathematical Monographs, (1990).

[34]

S. Lang, Introduction to Complex Hyperbolic Spaces,, Springer-Verlag, (1987). doi: 10.1007/978-1-4757-1945-1.

[35]

H. Liu, A Plane Rational Map with Chebyshev-like Dynamics,, Ph.D. thesis, (2014).

[36]

W. Parry, Entropy and Generators in Ergodic Theory,, W. A. Benjamin, (1969).

[37]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods,, London Mathematical Society Lecture Note Series, (2010). doi: 10.1017/CBO9781139193184.

[38]

C. Sabot, Spectral properties of self-similar lattices and iteration of rational maps,, Mém. Soc. Math. Fr. (N.S.), (2003).

[39]

G. Vigny, Hyperbolic measure of maximal entropy for generic rational maps of $\mathbbP^k$,, Ann. Inst. Fourier (Grenoble), 64 (2014), 645. doi: 10.5802/aif.2861.

show all references

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.

[2]

L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets,, in Fractals in Graz 2001, (2001), 25.

[3]

E. Bedford, S. Cantat and K. Kim, Pseudo-automorphisms with no invariant foliation,, J. Mod. Dyn., 8 (2014), 221. doi: 10.3934/jmd.2014.8.221.

[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. doi: 10.1007/BF01232426.

[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. doi: 10.1007/s10240-001-8190-4.

[6]

S. Cantat, Dynamique des automorphismes des surfaces $K3$,, Acta Math., 187 (2001), 1. doi: 10.1007/BF02392831.

[7]

S. Daurat, On the size of attractors in $\mathbbP^k$,, Math. Z., 277 (2014), 629. doi: 10.1007/s00209-013-1269-z.

[8]

H. De Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps,, Mém. Soc. Math. Fr. (N.S.), (2010).

[9]

J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces,, Amer. J. Math., 123 (2001), 1135. doi: 10.1353/ajm.2001.0038.

[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. doi: 10.1512/iumj.2010.59.4023.

[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.

[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. doi: 10.4171/CMH/224.

[13]

J. Diller and J.-L. Lin, Rational surface maps with invariant meromorphic two-forms,, Math. Ann., 364 (2016), 313. doi: 10.1007/s00208-015-1211-2.

[14]

T.-C. Dinh, Attracting current and equilibrium measure for attractors on $\mathbbP^k$,, J. Geom. Anal., 17 (2007), 227. doi: 10.1007/BF02930722.

[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.

[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. doi: 10.4007/annals.2005.161.1637.

[17]

R. Dujardin, Hênon-like mappings in $\mathbbC^2$,, Amer. J. Math., 126 (2004), 439.

[18]

C. Favre and J. V. Pereira, Foliations invariant by rational maps,, Math. Z., 268 (2011), 753. doi: 10.1007/s00209-010-0693-6.

[19]

J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension. II,, in Modern Methods in Complex Analysis (Princeton, (1992), 135.

[20]

J. E. Fornaess and N. Sibony, Dynamics of $P^2$ (examples),, in Laminations and Foliations in Dynamics, (1998), 47. doi: 10.1090/conm/269/04329.

[21]

J. E. Fornaess and B. Weickert, Attractors in $P^2$,, in Several Complex Variables (Berkeley, (1999), 1995.

[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. doi: 10.1023/A:1012061801279.

[23]

M. Gromov, Entropy, homology and semialgebraic geometry,, Séminaire Bourbaki, (1987), 145.

[24]

V. Guedj, Entropie topologique des applications méromorphes,, Ergodic Theory Dynam. Systems, 25 (2005), 1847. doi: 10.1017/S0143385705000192.

[25]

V. Guedj, Ergodic properties of rational mappings with large topological degree,, Ann. of Math. (2), 161 (2005), 1589. doi: 10.4007/annals.2005.161.1589.

[26]

V. Guedj, Propriétés ergodiques des applications rationnelles,, in Quelques aspects des systèmes dynamiques polynomiaux, (2010), 97.

[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), 89.

[28]

M. Jonsson, Hyperbolic dynamics of endomorphisms,, Unpublished note: , ().

[29]

M. Jonsson and B. Weickert, A nonalgebraic attractor in $\mathbbP^2$,, Proc. Amer. Math. Soc., 128 (2000), 2999. doi: 10.1090/S0002-9939-00-05529-5.

[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.

[31]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187.

[32]

B. P. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts,, Universitext, (1998). doi: 10.1007/978-3-642-58822-8.

[33]

S. G. Krantz, Complex Analysis: The Geometric Viewpoint,, Carus Mathematical Monographs, (1990).

[34]

S. Lang, Introduction to Complex Hyperbolic Spaces,, Springer-Verlag, (1987). doi: 10.1007/978-1-4757-1945-1.

[35]

H. Liu, A Plane Rational Map with Chebyshev-like Dynamics,, Ph.D. thesis, (2014).

[36]

W. Parry, Entropy and Generators in Ergodic Theory,, W. A. Benjamin, (1969).

[37]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods,, London Mathematical Society Lecture Note Series, (2010). doi: 10.1017/CBO9781139193184.

[38]

C. Sabot, Spectral properties of self-similar lattices and iteration of rational maps,, Mém. Soc. Math. Fr. (N.S.), (2003).

[39]

G. Vigny, Hyperbolic measure of maximal entropy for generic rational maps of $\mathbbP^k$,, Ann. Inst. Fourier (Grenoble), 64 (2014), 645. doi: 10.5802/aif.2861.

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