Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEntropy of meromorphic maps acting on analytic sets
94
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $f : Xto X$ be a dominating meromorphic map on a compact Kahler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0leq l leq k$. For an ergodic probability measure $ u$, we extend similarly the notion of measure-theoretic entropy $h_{ u}^l(f)$. Under mild hypothesis, we compute $h^l_{mathrm{top}}(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of $mathbb{P}^2$ of degree $d$, we show that $h^1_{mathrm{top}}(f)= log d$ for a large class of maps but we give examples where $h^1_{mathrm{top}}(f) eq log d$.
rate research
Read More
We study the dynamics of meromorphic maps for a compact Kaehler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of P^k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic.
This paper studies the uniqueness of two non-integral finite ordered meromorphic functions with finitely many poles when they share two finite sets. Also, studies an answer to a question posed by Gross for a particular class of meromorphic functions. Moreover, some observations are made on some results due to Sahoo and Karmakar ( Acta Univ. Sapientiae, Mathematica, DOI: 10.2478/ausm-2018-0025) and Sahoo and Sarkar (Bol. Soc. Mat. Mex., DOI: 10.1007/s40590-019-00260-4).
If $X$ is the attractor set of a conformal IFS in dimension two or three, we prove that there exists a quasiregular semigroup $G$ with Julia set equal to $X$. We also show that in dimension two, with a further assumption similar to the open set condition, the same result can be achieved with a semigroup generated by one element. Consequently, in this case the attractor set is quasiconformally equivalent to the Julia set of a rational map.
We consider a subanalytic subset A of a complex analytic manifold M (when M is viewed as a real manifold) and formulate conditions under which A is a complex analytic subset of M.
Let $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {sl semigroup-fication}, which allows to construct a continuous semigroup $(phi_t)$ of holomorphic self-maps of the unit disc whose time one map $phi_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of $f$ with a given multiplier and the corresponding set for $phi_1$. Moreover, in case $f$ (and hence $phi_1$) has no interior fixed points, the slope of the orbits converging to the Denjoy-Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of $f$ converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs domain of $f$ is almost symmetric with respect to vertical lines.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
