E. Amerik, F. Bogomolov, and M. Rovinsky, Abstract, Compositio Mathematica, vol.1, issue.06, pp.1819-1842, 2011.
DOI : 10.1090/S1056-3911-02-00328-4

E. Amerik, Existence of non-preperiodic algebraic points for a rational self-map of infinite order, Mathematical Research Letters, vol.18, issue.2, pp.251-256, 2011.
DOI : 10.4310/MRL.2011.v18.n2.a5

E. Amerik and F. Campana, Fibrations méromorphes sur certaines variétés variétésà fibré canonique trivial, Pure Appl. Math. Q, issue.2 1, pp.509-545, 2008.

M. Baker and L. Demarco, Special curves and postcritically-finite polynomials. to appear in Forum of Math

E. Bedford, M. Lyubich, and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of C2, Inventiones Mathematicae, vol.10, issue.1, pp.277-288, 1993.
DOI : 10.1007/BF01232671

J. P. Bell, D. Ghioca, and T. J. Tucker, Applications of p-adic analysis for bounding periods of subvarieties underétaleunderétale maps

J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell-Lang problem forétaleforétale maps, Amer. J. Math, vol.132, issue.6, pp.1655-1675, 2010.

R. L. Benedetto, D. Ghioca, P. Kurlberg, and T. J. Tucker, A case of the dynamical Mordell???Lang conjecture, Mathematische Annalen, vol.147, issue.1, pp.1-26, 2012.
DOI : 10.1007/s00208-010-0621-4

S. Boucksom, C. Favre, and M. Jonsson, Degree growth of meromorphic surface maps, Duke Mathematical Journal, vol.141, issue.3, pp.519-538, 2008.
DOI : 10.1215/00127094-2007-004

J. Briend and J. Duval, Deux caract??risations de la mesure d'??quilibre d'un endomorphisme de P k (C), Publications math??matiques de l'IH??S, vol.93, issue.1, pp.145-159, 2001.
DOI : 10.1007/s10240-001-8190-4

G. Call and J. H. Silverman, Canonical heights on varieties with morphisms, Compositio Math, vol.89, pp.163-205, 1993.

S. Cantat, Invariant Hypersurfaces in holomorphic Dynamics, Mathematical Research Letters, vol.17, issue.5, pp.833-841, 2010.
DOI : 10.4310/MRL.2010.v17.n5.a3
URL : https://hal.archives-ouvertes.fr/hal-00559935

S. Cantat, Sur les groupes de transformations birationnelles des surfaces, Annals of Mathematics, vol.174, issue.1, pp.299-340, 2011.
DOI : 10.4007/annals.2011.174.1.8
URL : https://hal.archives-ouvertes.fr/hal-00913863

S. Cantat and S. Lamy, Normal subgroups in the Cremona group, Acta Mathematica, vol.210, issue.1, pp.31-94, 2013.
DOI : 10.1007/s11511-013-0090-1
URL : https://hal.archives-ouvertes.fr/hal-00705299

S. D. Cutkosky and V. Srinivas, On a Problem of Zariski on Dimensions of Linear Systems, The Annals of Mathematics, vol.137, issue.3
DOI : 10.2307/2946531

T. Dinh and N. Sibony, Density of positive closed currents and dynamics of Henon-type automorphisms of C k (part II)

T. Dinh and N. Sibony, Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings, Holomorphic dynamical systems, pp.165-294, 1998.
DOI : 10.1007/978-3-642-13171-4_4

N. Fakhruddin, Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc, vol.18, issue.2, pp.109-122, 2003.

G. Faltings, The General Case of S. Lang's Conjecture, Barsotti Symposium in Algebraic Geometry, pp.175-182, 1991.
DOI : 10.1016/B978-0-12-197270-7.50012-7

C. Favre, Le groupe de Cremona et ses sous-groupes de type fini, Astérisque, vol.332, issue.998, pp.11-43, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00836025

C. Favre and M. Jonsson, The valuative tree, Lecture Notes in Mathematics, vol.1853, 2004.
DOI : 10.1007/b100262

C. Favre and M. Jonsson, Eigenvaluations, Annales Scientifiques de l?????cole Normale Sup??rieure, vol.40, issue.2, pp.309-349, 2007.
DOI : 10.1016/j.ansens.2007.01.002

C. Favre and M. Jonsson, Dynamical compactifications of $\C^2$, Annals of Mathematics, vol.173, issue.1, pp.211-248, 2011.
DOI : 10.4007/annals.2011.173.1.6
URL : https://hal.archives-ouvertes.fr/hal-00836011

D. Ghioca and T. J. Tucker, Periodic points, linearizing maps, and the dynamical Mordell???Lang problem, Journal of Number Theory, vol.129, issue.6, pp.1392-1403, 2009.
DOI : 10.1016/j.jnt.2008.09.014

D. Ghioca, T. J. Tucker, and S. Zhang, Towards a Dynamical Manin-Mumford Conjecture, International Mathematics Research Notices, vol.22, pp.5109-5122, 2011.
DOI : 10.1093/imrn/rnq283

D. Ghioca, T. J. Tucker, and M. E. Zieve, Intersections of polynomial orbits, and a dynamical Mordell???Lang conjecture, Inventiones mathematicae, vol.10, issue.2, pp.463-483, 2008.
DOI : 10.1007/s00222-007-0087-5

V. Guedj, Ergodic properties of rational mappings with large topological degree, Annals of Mathematics, vol.161, issue.3, pp.1589-1607, 2005.
DOI : 10.4007/annals.2005.161.1589

E. Hrushovski, The elementary theory of the Frobenius automorphisms. arX- iv:math/0406514v1

S. Kawaguchi, Local and global canonical height functions for affine space regular automorphisms, Algebra & Number Theory, vol.7, issue.5, pp.1225-1252, 2013.
DOI : 10.2140/ant.2013.7.1225

C. Lech, A note on recurring series, Arkiv f??r Matematik, vol.2, issue.5, pp.417-421, 1953.
DOI : 10.1007/BF02590997

C. G. Lee, An upper bound for height for regular affine automorphisms on A n

E. Wulcan and M. Jonsson, Canonical heights for plane polynomial maps of small topological degree

Y. I. Manin, Cubic forms Algebra, geometry, arithmetic, 1986.

A. Medvdev and T. Scanlon, Invariant varieties for polynomial dynamical systems. to appear in Ann, Math, issue.2

F. Pazuki, Zhang's conjecture and squares of Abelian surfaces, Comptes Rendus Mathematique, vol.348, issue.9-10, pp.483-486, 2010.
DOI : 10.1016/j.crma.2010.03.014

M. Raynaud, Courbes sur une vari???t??? ab???lienne et points de torsion, Inventiones Mathematicae, vol.44, issue.1, pp.207-233, 1983.
DOI : 10.1007/BF01393342

M. Raynaud, Sous-variétés dune variété abélienne et points de torsion. In Arithmetic and geometry, volume I of Progr, Math. Birkhäuser Boston, pp.327-352, 1983.

J. H. Silverman, Rational points on K3 surfaces: A new canonical height, Inventiones Mathematicae, vol.198, issue.1, pp.347-373, 1991.
DOI : 10.1007/BF01232270

J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol.241, 2007.
DOI : 10.1007/978-0-387-69904-2

J. H. Silverman, Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space. Ergodic Theory and Dynamical Systems, pp.647-678, 2014.

H. Joseph, J. Silverman, and . Felipe-voloch, A local-global criterion for dynamics on P 1, Acta Arithmetica, vol.137, issue.3, pp.285-294, 2009.

T. Dinh and V. Nguyen, Equidistribution for meromorphic maps with dominant topological degree, Indiana University Mathematics Journal, vol.64, issue.6
DOI : 10.1512/iumj.2015.64.5674

P. Vojta, Integral points on subvarieties of semiabelian varieties, I, Inventiones Mathematicae, vol.126, issue.1, pp.133-181, 1996.
DOI : 10.1007/s002220050092

M. Wibmer, Skolem???Mahler???Lech type theorems and Picard???Vessiot theory, Journal of the European Mathematical Society, vol.17, issue.3
DOI : 10.4171/JEMS/509

. Shreeram-shankar-abhyankar, Resolution of singularities of embedded algebraic surfaces, Pure and Applied Mathematics, vol.24, 1966.

E. Amerik, Existence of non-preperiodic algebraic points for a rational self-map of infinite order, Mathematical Research Letters, vol.18, issue.2, pp.251-256, 2011.
DOI : 10.4310/MRL.2011.v18.n2.a5

E. Bedford and J. Diller, Energy and invariant measures for birational surface maps. Duke Math, J, vol.128, issue.2, pp.331-368, 2005.

S. Boucksom, C. Favre, and M. Jonsson, Degree growth of meromorphic surface maps, Duke Mathematical Journal, vol.141, issue.3, pp.519-538, 2008.
DOI : 10.1215/00127094-2007-004

F. E. Brochero-martínez, F. Cano, and L. López-hernanz, Parabolic curves for diffeomorphisms in $(\mathbb{C}^2,0)$, Publicacions Matem??tiques, vol.52, issue.1, pp.189-194, 2008.
DOI : 10.5565/PUBLMAT_52108_09

S. Cantat, Sur la dynamique du groupe d'automorphismes des surfaces K3, Transformation Groups, vol.189, issue.No. 4, pp.201-214, 2001.
DOI : 10.1007/BF01263089
URL : https://hal.archives-ouvertes.fr/hal-00725118

S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Mathematica, vol.187, issue.1, pp.1-57, 2001.
DOI : 10.1007/BF02392831
URL : https://hal.archives-ouvertes.fr/hal-00725118

S. Cantat and S. Lamy, Normal subgroups in the Cremona group, Acta Mathematica, vol.210, issue.1, pp.31-94, 2013.
DOI : 10.1007/s11511-013-0090-1
URL : https://hal.archives-ouvertes.fr/hal-00705299

D. Cerveau and J. Déserti, Transformations birationnelles de petit degré, of Cours Spécialisés [Specialized Courses]. Société Mathématique de France

M. Mella, C. Bisi, and A. Calabri, On plane cremona transformations of fixed degree

M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol.1441, 1990.
DOI : 10.1007/BFb0084913

J. Déserti and D. Cerveau, Transformations birationnelles de petit degré. ´ A para??trepara??tre dans Cours Spécialisés

O. Debarre, Higher-dimensional algebraic geometry. Universitext, 2001.

J. Diller, R. Dujardin, and V. Guedj, Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory, Annales scientifiques de l'??cole normale sup??rieure, vol.43, issue.2
DOI : 10.24033/asens.2120
URL : https://hal.archives-ouvertes.fr/hal-00665974

J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, American Journal of Mathematics, vol.123, issue.6, pp.1135-1169, 2001.
DOI : 10.1353/ajm.2001.0038

J. Diller, D. Jackson, and A. Sommese, Invariant curves for birational surface maps, Trans. Amer. Math. Soc, vol.359, issue.6, pp.2793-2991, 2007.

T. Dinh and N. Sibony, Une borne sup??rieure pour l???entropie topologique d???une application rationnelle, Annals of Mathematics, vol.161, issue.3, pp.1611637-1644, 2005.
DOI : 10.4007/annals.2005.161.1637

R. Dujardin, Laminar currents and birational dynamics, Duke Mathematical Journal, vol.131, issue.2, pp.219-247, 2006.
DOI : 10.1215/S0012-7094-06-13122-8

N. Fakhruddin, Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc, vol.18, issue.2, pp.109-122, 2003.

C. Favre, Le groupe de Cremona et ses sous-groupes de type fini, Astérisque, vol.332, issue.998, pp.11-43, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00836025

J. E. , F. , and N. Sibony, Complex dynamics in higher dimension. II, Ann. of Math. Stud. Princeton Univ. Press, vol.137, 1995.

W. Fulton, Intersection theory, Mathematics and Related Areas Sur les groupes hyperboliques d'après Mikhael Gromov Papers from the Swiss Seminar on Hyperbolic Groups, 1984.

M. H. Gizatullin, Rational G-surfaces, Izv. Akad. Nauk SSSR Ser. Mat, vol.44, issue.1, pp.110-144, 1980.

R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol.52, issue.52, 1977.
DOI : 10.1007/978-1-4757-3849-0

E. Hrushovski, The elementary theory of the Frobenius automorphisms. arX- iv:math/0406514v1

K. Iwasaki and T. Uehara, Periodic points for area-preserving birational maps of surfaces, Mathematische Zeitschrift, vol.337, issue.2, pp.289-318, 2010.
DOI : 10.1007/s00209-009-0570-3

L. Steven and . Kleiman, Toward a numerical theory of ampleness, Ann. of Math, vol.84, issue.2, pp.293-344, 1966.

R. Lazarsfeld, Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics Classical setting: line bundles and linear series, Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, 2004.

M. Lejeune-jalabert and B. Teissier, Cl??ture int??grale des id??aux et ??quisingularit??, Annales de la facult?? des sciences de Toulouse Math??matiques, vol.17, issue.4, pp.781-859, 2008.
DOI : 10.5802/afst.1203

Y. I. Manin, Cubic forms Algebra, geometry, arithmetic, 1986.

S. Saito, General Fixed Point Formula for an Algebraic Surface and the Theory of Swan Representations for Two-Dimensional Local Rings, American Journal of Mathematics, vol.109, issue.6, pp.1009-1042, 1987.
DOI : 10.2307/2374584

J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol.106, 1986.

S. Zhang, Distributions in algebraic dynamics, surveys in differential geometry, pp.381-430, 2006.

. Shreeram-shankar-abhyankar, Resolution of singularities of embedded algebraic surfaces, Pure and Applied Mathematics, vol.24, 1966.

J. P. Bell, A GENERALISED SKOLEM???MAHLER???LECH THEOREM FOR AFFINE VARIETIES, Journal of the London Mathematical Society, vol.73, issue.02, pp.367-379, 2006.
DOI : 10.1112/S002461070602268X

J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell-Lang problem forétaleforétale maps, Amer. J. Math, vol.132, issue.6, pp.1655-1675, 2010.

R. L. Benedetto, D. Ghioca, P. Kurlberg, and T. J. Tucker, A case of the dynamical Mordell???Lang conjecture, Mathematische Annalen, vol.147, issue.1, pp.1-26, 2012.
DOI : 10.1007/s00208-010-0621-4

O. Debarre, Higher-dimensional algebraic geometry. Universitext, 2001.

L. Denis, G??om??trie et suites r??currentes, Bulletin de la Société mathématique de France, vol.122, issue.1, pp.13-27, 1994.
DOI : 10.24033/bsmf.2221

J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, American Journal of Mathematics, vol.123, issue.6, pp.1135-1169, 2001.
DOI : 10.1353/ajm.2001.0038

G. Faltings, The General Case of S. Lang's Conjecture, Barsotti Symposium in Algebraic Geometry, pp.175-182, 1991.
DOI : 10.1016/B978-0-12-197270-7.50012-7

S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorpautomo, Ergodic Theory Dyn. Syst, vol.9, pp.67-99, 1989.

W. Fulton, Algebraic curves. An introduction to algebraic geometry, Notes written with the collaboration of Richard Weiss, Mathematics Lecture Notes Series, 1969.

R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol.52, issue.52, 1977.
DOI : 10.1007/978-1-4757-3849-0

E. Kunz, Introduction to plane algebraic curves, Birkhäuser Boston Inc, 2005.

A. Laface and M. Velasco, A survey on Cox rings, Geometriae Dedicata, vol.316, issue.2, pp.269-287, 2009.
DOI : 10.1007/s10711-008-9329-y

J. Serre, Lectures on the Mordell-Weil theorem. Friedr, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, 1989.

J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol.106, 1986.

J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol.241, 2007.
DOI : 10.1007/978-0-387-69904-2

P. Vojta, Integral points on subvarieties of semiabelian varieties, I, Inventiones Mathematicae, vol.126, issue.1, pp.133-181, 1996.
DOI : 10.1007/s002220050092

T. When-n, There exists an increasing sequence v n ? [? deg, w] tending to w satisfying ?(v n ) ? lim v<w,v?w ?(v) = ?(w) Since w is an end point, we may suppose that T n := T \ (v n , w] is a finite tree There exists a function g ? SH + (V ? ) such that Supp ?g ? [? deg, w] and it is strict decreasing on [? deg, w]. By replacing ? i by ? i + g for i = 1, 2, we may suppose that ? i 's are strict decreasing on [? deg, w]. When ?(v n ) = ?(w), set ? n := R Tn ?, When ?(v n ) > ?(w), the function ? 1 (v) ? ? 2 (v n ) is decreasing. Observe that ? 1 (v m )?? 2 (v n ) = ?(v m )?? 2 (v n )+? 2 (v m ) ? ?(w)?? 2 (v n )+? 2 (w) when m ? ?

G. Faltings, The General Case of S. Lang's Conjecture, Barsotti Symposium in Algebraic Geometry, pp.175-182, 1991.
DOI : 10.1016/B978-0-12-197270-7.50012-7

C. Favre and M. Jonsson, The valuative tree, Lecture Notes in Mathematics, vol.1853, 2004.
DOI : 10.1007/b100262

C. Favre and M. Jonsson, Eigenvaluations, Annales Scientifiques de l?????cole Normale Sup??rieure, vol.40, issue.2, pp.309-349, 2007.
DOI : 10.1016/j.ansens.2007.01.002

M. Jonsson, Dynamics on Berkovich Spaces in Low Dimensions
DOI : 10.1007/978-3-319-11029-5_6

P. Mondal, Analytic compactifications of C 2 part I -curvettes at infinity. arX- iv:1110, p.6905

P. Mondal, An effective criterion for algebraic contractibility of rational curves. arX- iv:1301

P. Mondal, How to determine the sign of a valuation on C[x, y]

S. Najib, Factorisation des polyômiales P (x 1 , · · · , x n ) ? ? et Théorème de Stein, 2005.

S. Schröer, On contractible curves on normal surfaces, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.2000, issue.524, pp.1-15, 2000.
DOI : 10.1515/crll.2000.057

J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol.106, 1986.

T. T. Mok, On analytic irreducibility at ? of a pencil of curves, Proc. Amer, pp.22-24, 1974.

J. Xie, The Dynamical Mordell-Lang Conjecture for polynomial maps on affine plane