Physical measures for partially hyperbolic surface endomorphisms
Physical measures for partially hyperbolic surface endomorphisms
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DOI：
10.1007/bf02392516
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发表时间：
2003-01
期刊：
影响因子：
3.7
通讯作者：
M. Tsujii
中科院分区：
文献类型：
--
作者：
M. Tsujii
在从遍历理论的观点研究光滑动力系统时，最基本的问题之一是下面的较佳图景是否对几乎所有的光滑动力系统都成立：勒贝格轨道的渐近分布几乎每个初始点都存在，并且与给出的动力系统的有限多个遍历不变度量之一重合。一般来说，答案是肯定的[14]。然而，在一般情况下回答这个问题似乎远远超出了目前的研究范围。设M是二维环面T=R2/Z2，或者更一般地，环面T上的一个区域，其边界由有限多条简单的闭C2-曲线组成：例如环(R/Z)x[-89 1].我们用欧几里德空间R2上的标准度量明显诱导出的黎曼度量[]-[]和勒贝格度量m来装备M。如果有正的常数A和c，且切丛Tm-E c Ge u有dim E e=dim E u=1的连续分解，我们称C l-映射F：M-+M为部分双曲自同态.
In the study of smooth dynamical systems from the standpoint of ergodic theory, one of the most fundamental questions is whether the following preferable picture is true for almost all of them: The asymptotic distribution of the orbit for Lebesgue almost every initial point exists and coincides with one of the finitely many ergodic invariant measures that are given for the dynamical system. The answer is expected to be affirmative in general [14]. However, it seems far beyond the scope of present research to answer the question in the general setting. The purpose of this paper is to provide an affirmative answer to the question in the case of partially hyperbolic endomorphisms on surfaces with one-dimensional unstable subbundle.Let M be the two-dimensional torus T= R2/Z 2 or, more generally, a region on the torus T whose boundary consists of finitely many simple closed C2-curves: eg an annulus (R/Z) x [-89 1]. We equip M with the Riemannian metric []-[] and the Lebesgue measure m that are induced by the standard ones on the Euclidean space R 2 in an obvious manner. We call a Cl-mapping F: M--+ M a partially hyperbolic endomorphism if there are positive constants A and c and a continuous decomposition of the tangent bundle TM--E c GE u with dim E e= dim E u= 1 such that