论文标题
开放缩放系统的平衡状态
Equilibrium States for Open Zooming Systems
论文作者
论文摘要
在这项工作中，我们为适合特殊类型孔的缩放系统构建了马尔可夫结构。我们的施工基于缩放时间提供的向后收缩。这些马尔可夫结构可用于编码开放的缩放系统。 在开放的缩放系统的背景下，可能在临界/奇异集合的情况下，我们证明存在有限的许多千古化变焦平衡状态，用于变焦电位，其诱导潜力是本地Hölder。例如，缩放Hölder连续。在变焦中，有所谓的\ textit {双曲电势}，也是我们所说的\ textit {有界的失真电位}，作为一种特殊情况，是\ textit {pseudo-deomements} $ ϕ $ ϕ_ {t} = -t \ logj_μf$ jacuian，$ j__ y y是jac $ jac $ jac $ jac y是jac的jac y;此外，对于最后一类潜力，我们展示了我们所谓的伪符号措施的存在。之后，由于条件温和，我们证明了平衡状态的唯一性，无需传递性。 该技术包括在有限的马尔可夫结构中使用诱导方案，该结构具有无限的许多符号来编码动力学，以获得相关符号动力学的平衡状态，然后将其投影以获得原始映射的平衡状态。为了获得伪符号措施，我们“传播”了用于符号动力学的共形量度。通过表明平衡状态可将相同的诱导方案提升来获得唯一性。 最后，我们表明双曲线电位类别等于连续变焦电位的类别。此外，我们给出了一类双曲电势的例子（包括无效的电位）。它意味着平衡状态的存在和独特性。在所考虑的地图中，有一个称为Viana地图的重要类。
In this work, we construct Markov structures for zooming systems adapted to holes of a special type. Our construction is based on backward contractions provided by zooming times. These Markov structures may be used to code the open zooming systems. In the context of open zooming systems, possibly with the presence of a critical/singular set, we prove the existence of finitely many ergodic zooming equilibrium states for zooming potentials whose induced potential is locally Hölder. For example, the zooming Hölder continuous. Among the zooming ones are the so-called \textit{hyperbolic potentials} and also what we call \textit{bounded distortion potentials}, having as a particular case the \textit{pseudo-geometric potentials} $ϕ_{t} = -t \log J_μf $, where $J_μf$ is a Jacobian of the reference zooming measure. Moreover, for this last class of potentials, we show the existence of what we call pseudo-conformal measures. Afterwards, with a mild condition, we prove uniqueness of equilibrium state with no requirement of transitivity. The technique consists in using an inducing scheme in a finite Markov structure with infinitely many symbols to code the dynamics to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. To obtain a pseudo-conformal measure, we "spread" the conformal one which exists for the symbolic dynamics. The uniqueness is obtained by showing that the equilibrium states are liftable to the same inducing scheme. Finally, we show that the class of hyperbolic potentials is equivalent to the class of continuous zooming potentials. Moreover, we give a class of examples of hyperbolic potentials (including the null one). It implies the existence and uniqueness of equilibrium state. Among the maps considered is the important class known as Viana maps.
