论文标题
缩放系统的亚酒精学和活泼定理
Subcohomology and a Livsic Theorem for Zooming Systems
论文作者
论文摘要
在连续缩放系统的背景下，紧凑型公制$ m $上的$ f：m \至m $，其中包括非均匀扩展的膨胀，可能存在关键集合，并在$ m $中具有缩放套件，我们证明，任何hölder$ ϕ：m \ to for \ mathd $ for \ m ys for \ mathd $ for \ mathd $ for \ math. $ f $ -invariant概率$μ$，承认连续函数$λ_{0}：m \ to \ mathbb {r} $（如果某些积分是正面的，可以是hölder） \ [ ϕ \geqλ_{0}-λ_{0} \ circ f。 \] 对于$ c^{1} $ - 在圆圈上扩展图$ \ mathbb {t} = \ Mathbb {r}/\ mathbb {z} $在[9]中扩展了一个结果，以均匀扩展，局部差异性扩张，不均匀，viana maps and benedicks-car and andecare maps and ned n. mmaps and nedicks-carononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononemononemononemmos car。我们还举例说明了指数收缩的情况。 此外，对于积分$ \ int ϕdμ = 0 $，相对于任何$ f $ invariant的概率$μ$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ univsic $λ_{0} $，我们可以获得一个livsic定理的定期点。 \ [ ϕ =λ_{0}-λ_{0} \ circ f。 \] 此外，我们还证明，最大化积分的度量对于残余电势是唯一的。
In the context of continuous zooming systems $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, with the zooming set dense in $M$, we prove that any Hölder potential $ϕ: M \to \mathbb{R}$ for which the integrals $\int ϕdμ\geq 0$ with respect to any $f$-invariant probability $μ$, admits a continuous function $λ_{0} : M \to \mathbb{R}$ (which can be Hölder if some integral is positive) such that \[ ϕ\geq λ_{0}- λ_{0} \circ f. \] This extends a result in [9] for $C^{1}$-expanding maps on the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ to important classes of maps as uniformly expanding, local diffeomorphisms with non-uniform expansion, Viana maps, Benedicks-Carleson maps and Rovella maps. We also give an example beyond the exponential contractions context. Moreover, in the case of the integrals $\int ϕdμ= 0$ with respect to any $f$-invariant probability $μ$ and the set of periodic points to be dense in $M$, we obtain a version of the Livsic Theorem, that is, the functions $λ_{0}$ can be taken such that \[ ϕ= λ_{0}- λ_{0} \circ f. \] Additionally, we also prove that the measure which maximizes the integrals is unique for a residual set of potentials.
