Une borne supérieure pour l’entropie topologique d’une application rationnelle

Dinh, Tien-Cuong; Sibony, Nessim

doi:10.4007/annals.2005.161.1637

Cited by 157 publications

(201 citation statements)

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“…On the former side, algebraic geometry of Painlevé equations, especially a moduli-theoretical formulation of Painlevé dynamical systems [16,17] is an essential ingredient of our discussion. On the latter side, recent advances in complex surface dynamics, especially some deep ergodic studies of birational maps of complex surfaces [2,7,8,10] are another basis of our analysis. These two stuffs are combined fruitfully via a Riemann-Hilbert correspondence to reveal the chaotic nature of the sixth Painlevé dynamics.…”

Section: Introductionmentioning

confidence: 99%

An ergodic study of Painlevé VI

Iwasaki

Uehara

2007

Math. Ann.

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An ergodic study of Painlevé VI is developed. The chaotic nature of its Poincaré return map is established for almost all loops. The exponential growth of the numbers of periodic solutions is also shown. Principal ingredients of the arguments are a moduli-theoretical formulation of Painlevé VI, a Riemann-Hilbert correspondence, the dynamical system of a birational map on a cubic surface, and the Lefschetz fixed point formula.

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Section: Introductionmentioning

confidence: 99%

An ergodic study of Painlevé VI

Iwasaki

Uehara

2007

Math. Ann.

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“…p X g for some constant A > 0 independent of T . The following semi-regularization of currents was proved by Sibony and the first author in [6], [7]. We need the following lemma.…”

Section: Positive Closed Currentsmentioning

confidence: 99%

“…It was shown in [6], [7] that dynamical degrees are bi-meromorphic invariants, i.e. conjugate maps have the same dynamical degrees.…”

Section: Proposition 31 Let T Be a Positive Closed P; P/-current Amentioning

confidence: 99%

“…It was shown in [6], [7] that OE p .f n / 1=n converges to a constant d p .f / which is the dynamical degree of order p of f . Note that the main difficulty here is that in general we do not have .f nCs / D .f n / B .f s / on cohomology classes.…”

Section: Dynamical Degreesmentioning

confidence: 99%

“…This is also a positive closed .p; p/-current. We will use the properties that k .T /k Ä AkT k and k .S/k Ä AkSk for some constant A > 0 independent of T , S, see [6], [7] for details.…”

Section: Dynamical Degreesmentioning

confidence: 99%

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Comparison of dynamical degrees for semi-conjugate meromorphic maps

Dinh¹,

Nguyên²

2011

Comment. Math. Helv.

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Decay of correlations and the central limit theorem for meromorphic maps

Dinh

Sibony

2006

Comm Pure Appl Math

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Let f be a dominant meromorphic self-map of large topological degree on a compact Kähler manifold. We give a new construction of the equilibrium measure µ of f and prove that µ is exponentially mixing. As a consequence, we get the central limit theorem in particular for Hölder-continuous observables, but also for noncontinuous observables. DECAY OF CORRELATIONS 755THEOREM 1.1 The measure µ is exponentially mixing in the following sense: for every > 0 and 0 < α ≤ 1, there exists a constant A ,α > 0 such thatfor all n ≥ 0, for every function ψ ∈ L ∞ (µ), and for every Hölder-continuous function ϕ of order α. If a real-valued Hölder-continuous function ϕ is not a coboundary and verifies µ, ϕ = 0, then it satisfies the CLT theorem.We recall in Section 2 the Gordin-Liverani theorem and in Section 3 some properties of the Sobolev space W 1,2 . The space of test functions W 1,2 * and a subspace W 1,2 * * are introduced in Section 4. This is the key point of the method. The new space W 1,2 * seems to be useful for complex dynamics in several variables. In complex dimension 1 it coincides with the space W 1,2 . In higher dimensions it takes into account the fact that not all currents of bidegree (1, 1) are closed. It enjoys the composition properties under meromorphic maps, useful for a space of observables. In Section 5, we give the new construction of µ, and in Section 6 its statistical properties. Note that the geometric decay of correlations for Hénon maps was recently proved in [6] using the dd c -method. Like the dd c -method, the d-method can be applied to some meromorphic correspondences and to random iteration [7].Notation. We will use different subspaces of L 1 (X ). Most of them carry a canonical (quasi) norm. For the space C 0 (X ) of continuous functions we use the supnorm, for the space Lip(X ) of Lipschitz functions the norm · L 1 + · Lip where ϕ Lip := sup x =y |ϕ(x) − ϕ(y)| dist(x, y) −1 , and for the Hölder space C α (X ) the norm ϕ C α := ϕ L 1 + sup x =y |ϕ(x) − ϕ(y)| dist(x, y) −α . We use the (quasi)norm · E + · F for the intersection E ∩ F of E and F and write L p instead of L p (X ) when there is no confusion. The L p norm of a form is the sum of L p norms of its coefficients for a fixed atlas of X . The topology that we consider is a weak topology. That is, ϕ n → ϕ in E if ϕ n → ϕ in the sense of distributions and ( ϕ n E ) is bounded. The continuity of linear operators : E → F is with respect to these topologies of E and F. The inclusion map E ⊂ L 1 (X ) is always bounded for the associated (quasi)-norms and is continuous in our sense.

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Une borne supérieure pour l’entropie topologique d’une application rationnelle

Abstract: Let X be a complex projective manifold and let f be a dominating rational map from X onto X. We show that the topological entropy h(f ) of f is bounded from above by the logarithm of its maximal dynamical degree.

Cited by 157 publications

References 19 publications

An ergodic study of Painlevé VI

An ergodic study of Painlevé VI

Comparison of dynamical degrees for semi-conjugate meromorphic maps

Decay of correlations and the central limit theorem for meromorphic maps

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