Regularization of currents and entropy

By T. Dinh, N. Sibony. 2004

149 citations recorded by the source corpus.

Abstract Let T be a positive closed ( p , p ) -current on a compact Kahler manifold X. We prove the existence of smooth positive closed ( p , p ) -forms T n + and T n − such that T n + − T n − → T weakly. Moreover, ‖ T n ± ‖ ⩽ c X ‖ T ‖ where c X > 0 is a constant independent of T. We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed ( 1 , 1 ) -currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate for the topological entropy of meromorphic maps on compact Kahler manifolds.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3a81d13a669903f2d80147621ad133ea4c06165f

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Citations in the corpus

Comparison of dynamical degrees for semi-conjugate meromorphic maps (2009, background): “The last integral is equal to ap,p(n).”
Decay of correlations and the central limit theorem for meromorphic maps (2004, methodology): “w if smooth functions are dense in W1,2 ∗ nor in W1,2 ∗ ∩ C 0(X). Hence, we don’t know if the extension of µis unique and a priori we don’t have hν,ϕi∗ = hν,ϕi on W1,2 ∗ ∩ C 0(X). The method, used in [9] to regularize currents, might give the solution to this question. The proof of the following pro”
Super-potentials of positive closed currents, intersection theory and dynamics (2007, background): “ular to f−1 instead of f. Let λ p denote the intermediate degree of order p of f and λp(fn) the intermediate degree of order p of fn. Note that λ 1(f) = d. We have the following elementary lemma, see [21, 22] for a more general context. Lemma 5.3.1. The sequence of intermediate degrees λp(fn) is sub”
Density of positive closed currents, a theory of non-generic intersections (2012, result): “may diﬀer by singular currents and a good choices of potential depend on the choice of coordinates. We refer the reader to the works by Demailly, Fornæss, Lelong, M´eo, Siu, Skoda, Vigny, the authors [6, 14, 18, 19, 23, 29, 30, 31, 32, 33, 35] and the references therein for results and applications ”
Distribution des valeurs de transformations méromorphes et applications (2006, background): “Certains résultats énoncés alors dans le cas projectif sont étendus ici au cas kählérien grâce au théorème d’approximation des courants démontré par les auteurs dans [7].”
Rigidity of Julia sets for Henon type maps (2013, background): “One has to use the regularization method in [25] to find smooth positive (k − 2, k − 2)-forms Rn on X converging to a current R′ ≥ R such that ‖Rn‖ ≤ c‖R‖ for some constant c ≥ 0 depending only on (X,ω).”
Degrees of iterates of rational maps on normal projective varieties (2017, methodology): “When K = C, Theorem 1 was proved by Dinh and Sibony in [11], and further generalized to compact Kähler manifolds in [10].”
Equidistribution problems of complex dynamics in higher dimension (2016, background): “ndp,m which implies the result. The proof of the abovetheorem in the general case usesin an essential way a computation with positive closed currents and Theorem 3.1 plays a crucial role. We refer to [22, 26, 30, 74, 75] for details and some extensions of this result. We also obtained in these works”
Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy (2013, methodology): “us of f∗|H2k(M,Z). In particular, if f is of positive entropy, then f is of inﬁnite order. The theorem of Gromov and Yomdin was partially extended to the case of meromorphic maps in Dinh-Sibony [DS05][DS04]. The main aim of this note is to present ﬁrst explicit examples of a rational threefold and a”
Pull-back of currents by holomorphic maps (2006, background): “The assertion on {F (T )} follows from this continuity and a regularization of T , see [6, 11].”
Dynamics of automorphisms on projective complex manifolds (2008, background): “Results like Lemmas 2.20 and 2.23 are not so essentially used by the paper, but are hopefully of independent interest.”