对于任何$C\in[0,\infty]$一个紧群自同构$T:X\to X$构造的性质，$$ \frac{1}{n}\log|\{x\in X\mid T^n(x)=x\}|\longrightarrow C. $$这可以被解释为一个组合模拟的（仍然开放）的问题，是否紧群自同构存在与任何给定的拓扑熵。
英文摘要：
For any $C\in[0,\infty]$ a compact group automorphism $T:X\to X$ is constructed with the property that $$ \frac{1}{n}\log|\{x\in X\mid T^n(x)=x\}|\longrightarrow C. $$ This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological entropy.
We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of $\mathbb{C}^{d}$ admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent to a complex submanifold, and that every sta…
We prove that for a polynomial diffeomorphism of C^2, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular f satisfies Smale's Axiom A on C^2 .
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $\mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a si…
For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple, and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group.
Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in $S$, we give a complete description of triples $(f_1,f_2,p)$ such that $p$ is a rational periodic point for both $f_i\in S$, $i=1,2$. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two n…
On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties
Osaka Metropolitan University（大阪公立大学）；NTT Institute for Fundamental Mathematics, NTT Communication Science Laboratories, NTT, Inc.（日本电信电话株式会社基础数学研究所、通信科学实验室）
We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic points, and few saddle periodic points.
We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sof…
We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka-Stein manifold. In the former case we need to impose a new tameness condition. It follows that hyperbolic periodic points are dense in the tame non-wandering set of a generic automorphism of a Stein manifold with the density property and in the non-wandering set of a generic endomorphism of an Oka-Stein manifold. These are the first results about holomorphic dynamics on Oka manifolds. We strengthen previous results of ours on the existence and genericity of chaotic volume-pre…
