Growth of the number of periodic points for meromorphic maps

ABSTRACT. Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree. The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. In this paper, we will survey some results and tools developed during the last two decades. Related concepts, new techniques and open problems will be presented.Classification AMS 2010: 32H, 32U, 37F.

show abstract

“…We first have the following general result recently obtained in [24]. Theorem 5.1 (Dinh-Nguyen-Truong).…”

Section: Number Of Isolated Periodic Points and Equidistributionmentioning

confidence: 99%

Equidistribution problems in complex dynamics of higher dimension

Dinh

Sibony

2017

Int. J. Math.

View full text Add to dashboard Cite

show abstract

“…Clearly, if f is a polynomial map, the cardinality of Per 0 n f is bounded by the degree of f n , which grows at most exponentially fast (see the generalization [18]). The first study in the C ∞ -case goes back to Artin and Mazur [2], who proved the existence of a dense set D in Diff ∞ (M ) formed by diffeomorphisms f such that the cardinality of Per 0 n f grows at most exponentially fast:…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Generic family displaying robustly a fast growth of the number of periodic points

Berger

2021

Acta Mathematica

View full text Add to dashboard Cite

“…We note a parallel between the least negative intersection and the tangent currents in this situation. Under the same assumptions as in Theorem 4.2, it was shown in our joint paper [8] that the h-dimension (defined in [10]) between T + and T − is 0, the best possible.…”

Section: 2mentioning

confidence: 87%

Strong submeasures and applications to non-compact dynamical systems

Truong

2020

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

show abstract

Growth of the number of periodic points for meromorphic maps

Cited by 19 publications

References 58 publications

Equidistribution problems in complex dynamics of higher dimension

Equidistribution problems in complex dynamics of higher dimension

Generic family displaying robustly a fast growth of the number of periodic points

Strong submeasures and applications to non-compact dynamical systems

Contact Info

Product

Resources

About