A Smale space is a chaotic dynamical system with canonical coordinates of contracting and expanding directions. The basic sets for Smale's Axiom A systems are a key class of examples. We consider the special case of irreducible Smale spaces with zero dimensional contracting directions, and characterize these as stationary inverse limits satisfying certain conditions.
Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions…
Associative Memory and Generative Diffusion in the Zero-noise Limit
Computational & Systems Biology, Sloan Kettering Institute, Memorial Sloan Kettering Cancer Center（计算生物学与系统生物学，斯隆凯特灵研究所，纪念斯隆凯特灵癌症中心）；Tri-Institutional Training Program in Computational Biology & Medicine（计算生物学与医学三机构联合培训计划）
Among other results, the paper gives new mapping theorems and new fixed point property theorems for inverse limits of inverse sequences of compact metric spaces with upper semicontinuous set-valued bonding functions. We also revisit the results from two papers, Mioduszewski's ''Mappings of inverse limits'' and Feuerbacher's ''Mappings of inverse limits revisited''.
For a continuous function $f : [0,1] \to [0,1]$ we define a splitting sequence admitted by $f$ and show that the inverse limit of $f$ is an arc if and only if $f$ does not admit a splitting sequence.
Inverse limits and statistical properties for chaotic implicitly defined economic models
本文研究了一类隐式定义的经济模型的动力学和遍历论。我们考虑一维和二维世代重叠模型，现金预付模型，异质市场和蛛网模型的自适应调整。研究了一类混沌不变分形集的逆极限空间及其度量、遍历和稳定性。逆极限给出了所考虑的经济问题的跨期完美预见均衡集。首先，我们证明了这些模型的逆极限在扰动下是稳定的。我们证明了逆极限是可扩的，并且具有规范性质。然后，在我们的例子中，我们在逆极限上使用效用函数。我们给出了两种方法来排名这样的效用函数。首先，当扰动某些动力系统，我们排名效用函数的\textit{average values}方面的不变概率测度的逆极限，特别是相对于最大熵的措施。对于某些单峰映射族，我们可以调整折扣因子和系统参数，以获得最大的平均效用值。第二种排序效用函数的方法（对于双曲集上更一般的映射）是使用这些效用函数在逆极限上的平衡测度;它们优化效用函数的平均值，同时从长远来看，\textit{at the same time}使系统中的无序度尽可能低。
We give an upper bound for the topological entropy of maps on inverse limit spaces in terms of their set-valued components. In a special case of a diagonal map on the inverse limit space $\underleftarrow{\lim}(I,f)$, where every diagonal component is the same map $g\colon I\to I$ which strongly commutes with $f$ (i.e. $f^{-1}\circ g=g\circ f^{-1}$), we show that the entropy equals $\max\{\textrm{Ent}(f),\textrm{Ent}(g)\}$. As a side product, we develop some techniques for computing topological entropy of set-valued maps.
In this paper, we consider inverse limits of [0,1] using upper semicontinuous set-valued bonding functions with the intermediate value property. Expanding on classical results by Barge and Martin, we explore the relationship between periodicity in the bonding function and the topology of the corresponding inverse limit. In particular, for an inverse limit of a single upper semicontinuous bonding map with the intermediate value property, we provide sufficient conditions for the existence of a periodic cycle with period not a power of two in the bonding function to imply the existence of an inde…
In this paper, we consider inverse limits of $[0,1]$ using upper semicontinuous set-valued functions. We aim to expand on a previous paper exploring the relationship between the existence periodic points of a continuous function to the existence of indecomposable subcontinua of the corresponding inverse limit. In a previous paper, sufficient conditions were given such that if a satisfactory bonding map $F$ had a periodic cycle of period not a power of 2, then $\lim\limits_{\leftarrow}\{[0,1],F\}$ contains an indecomposable continuum. We show that the condition that $F$ is almost nonfissile is …
