Asymptotic distributions of preimages for endomorphisms
Asymptotic distributions of preimages for endomorphisms
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自同态原像的渐近分布
DOI：
10.1017/s0143385710000155
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发表时间：
2010
影响因子：
0.9
通讯作者：
Eugen Mihailescu
中科院分区：
文献类型：
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作者：
Eugen Mihailescu
摘要双曲微分同胚的吸引子具有独特的Sinai-Ruelle-Bowen测度和有趣的性质。本文研究了具有驱避器Λ的不可逆映射（自同态）的情况。我们使用驱避器附近点的预像（假设是非扩展的）；这里的情况不同于微分同态或者自同态的正迭代。给出了从局部逆迭代中获得不变测度的两种方法。我们证明了如果Λ是f的双曲s-共形驱避器，不一定展开，如果f在Λ上是d- 1，那么对于Lebesgue， Λ驱避盆中的几乎每个x都有x的渐近分布历史，类似于Hölder连续势Φs的平衡测量μs，对于y∈Λ， Φs (y):=log∣Dfs (y)∣。度量μs在非可逆排斥体上起逆Sinai-Ruelle-Bowen度量的作用。我们还证明了存在一个集合a∧Wuε（Λ），其中Λ (a)= Λ （Wuε（Λ））（其中Λ（⋅）是Lebesgue测度），使得对于任意z∈a和任意实连续函数g， \[ {\int }_{\!\!W^u_\varepsilon (\Lambda )} |\mu _n^z(g) - \mu _s(g)| \,d\lambda (z) \mathop {\to }\limits _n 0, \]与\[\mu _n^z := \frac 1n \sum _{k=1}^n \frac {\sum _{y \in f_\Lambda ^{-k} z} \delta _y}{d^k}. \]，我们得到了在？M, M≥2。
Abstract Attractors for hyperbolic diffeomorphisms are known to possess unique Sinai–Ruelle–Bowen measures with interesting properties. In this paper we investigate the case of non-invertible maps (endomorphisms) which have repellers Λ. We work with preimages of points in a neighbourhood of the repeller (assumed to be non-expanding); the situation here is different than the one for diffeomorphisms or positive iterates of endomorphisms. We give two methods to obtain invariant measures from local inverse iterates. We show that if Λ is a hyperbolic s-conformal repeller for f, not necessarily expanding, and if f is d-to-1 on Λ then for Lebesgue almost every x in the repelling basin of Λ there are histories of x asymptotically distributed like the equilibrium measure μs of the Hölder continuous potential Φs, with Φs (y):=log ∣Dfs (y)∣ for y∈Λ. The measure μs plays the role of an inverse Sinai–Ruelle–Bowen measure on the non-invertible repeller. We prove also that there exists a set A⊂Wuε(Λ) with λ(A)=λ(Wuε(Λ)) (where λ(⋅) is the Lebesgue measure) such that for any z∈A and any real continuous function g, \[ {\int }_{\!\!W^u_\varepsilon (\Lambda )} |\mu _n^z(g) - \mu _s(g)| \,d\lambda (z) \mathop {\to }\limits _n 0, \] with \[\mu _n^z := \frac 1n \sum _{k=1}^n \frac {\sum _{y \in f_\Lambda ^{-k} z} \delta _y}{d^k}. \] In particular, we obtain the asymptotic distribution of preimages of Lebesgue almost all points for a class of hyperbolic toral endomorphisms on ?m,m≥2 .