Comparison of  dynamical degrees for semi-conjugate   meromorphic  maps

Dinh, Tien-Cuong; Nguyên, Viêt-Anh

doi:10.4171/cmh/241

Cited by 56 publications

(93 citation statements)

References 17 publications

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“…In the algebraic setting, over fields K different from C, there is still no definition of cohomological entropy. There are indications [6,12,26] that one should be willing to work with objects such asétale topology -seemingly further away from dynamical systems -in order to come closer to satisfying Gromov -Yomdin 's theorem. In fact, for automorphisms of compact surfaces, the work in [12] relates the quantity in Gromov -Yomdin's theorem to l-adic cohomology, which suggests that we may even need to deal with stranger objects thanétale topology.…”

Section: Algebraicétale Dynamical Systems and Entropymentioning

confidence: 99%

“…Even when K = C, the fact that the dynamical degrees are invariant under generically finite semi-conjugacies ( [6], see [26] for a generalisation to correspondences) is an evidence in support of the philosophy that to understand the dynamics of a rational map f : X → X, it is useful and necessary to study all rational maps f ′ : X ′ → X ′ (or even correspondences) which are semi-conjugate to f via a generically finite rational map…”

Section: Algebraicétale Dynamical Systems and Entropymentioning

confidence: 99%

“…First, we consider the case where K = C. Let f : X → X be a dominant rational map or more generally dominant correspondence, where X is a complex projective. Let B(f ) be the set of all dominant correspondences f Z : Z → Z, where Z is a complex projective variety so that is in fact the same as that for each individual f Z , see [6,26]. We then define theétale analogue of entropy…”

Section: Algebraicétale Dynamical Systems and Entropymentioning

confidence: 99%

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Étale dynamical systems and topological entropy

Truong¹

2023

Proc. Amer. Math. Soc.

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Abstract. In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of usinǵ etale analogs of topological and algebraic dynamical systems.The first question is to define topological entropy for a topological dynamical system (f, X, Ω). The main idea is to make use -in addition to invariant compact subspaces of (X, Ω) -of compactifications ofétale covers• π and the fibers of π are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results.The second question is to define topological entropy for algebraic dynamical systems,with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop anétale analog of algebraic dynamical systems.

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Section: Algebraicétale Dynamical Systems and Entropymentioning

confidence: 99%

Section: Algebraicétale Dynamical Systems and Entropymentioning

confidence: 99%

Section: Algebraicétale Dynamical Systems and Entropymentioning

confidence: 99%

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Étale dynamical systems and topological entropy

Truong¹

2023

Proc. Amer. Math. Soc.

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“…[3]). In particular, |aF ′ | (a > 0) is composed of a pencil (necessarily parametrized by a curve B ∼ = P 1 because the irregularity q(B) ≤ q(X) = 0) stabilized by f w .…”

Section: Proof Of Theorems When P =mentioning

confidence: 99%

Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces

Perroni

Zhang

2013

Math. Ann.

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Abstract. We use a concise method to construct pseudo-automorphisms f n of the first dynamical degree d 1 (f n ) > 1 on the blowups of the projective n-space for all n ≥ 2 and more generally on the blowups of products of projective spaces. These f n , for n = 3 have positive entropy, and for n ≥ 4 seem to be the first examples of pseudo-automorphisms with d 1 (f n ) > 1 (and of non-product type) on rational varieties of higher dimensions.

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“…Primitivity of f can also be detected from dynamical degrees via the following criterion (see [36]), which is a consequence of results in [19] and [20]: If λ 1 (f ) = λ 2 (f ) then f is primitive.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of blowups of threefolds being Fano or having Picard number 1

Truong¹

2016

Ergod. Th. Dynam. Sys.

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Abstract. Let X 0 be a smooth projective threefold which is Fano or which has Picard number 1. Let π : X → X 0 be a finite composition of blowups along smooth centers. We show that for "almost all" of such X, if f ∈ Aut(X) then its first and second dynamical degrees are the same. We also construct many examples of finite blowups X → X 0 , on which any automorphism is of zero entropy.The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology classes.We will also discuss a possible application of these results to a threefold constructed by Kenji Ueno.

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Comparison of dynamical degrees for semi-conjugate meromorphic maps

Cited by 56 publications

References 17 publications

Étale dynamical systems and topological entropy

Étale dynamical systems and topological entropy

Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces

Automorphisms of blowups of threefolds being Fano or having Picard number 1

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