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Register a new userFinite mean dimesnion and marker property
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Authors
Ruxi Shi
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In this paper, we develop the theory of $mathbb{Z}_p$-index which has been introduced by Tsukamoto, Tsutaya and Yoshinaga. As an application, we show that given any positive number, there exists a dynamical system with mean dimension equal to such number such that it does not have the marker property.
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We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Luck rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.
In this paper, the existence conditions of nonuniform mean-square exponential dichotomy (NMS-ED) for a linear stochastic differential equation (SDE) are established. The difference of the conditions for the existence of a nonuniform dichotomy between an SDE and an ordinary differential equation (ODE) is that the first one needs an additional assumption, nonuniform Lyapunov matrix, to guarantee that the linear SDE can be transformed into a decoupled one, while the second does not. Therefore, the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity. This paper is also concerned with the mean-square exponential stability of nonlinear perturbation of a linear SDE under the condition of nonuniform mean-square exponential contraction (NMS-EC). For this purpose, the concept of second-moment regularity coefficient is introduced. This concept is essential in determining the stability of the perturbed equation, and hence we deduce the lower and upper bounds of this coefficient. Our results imply that the lower and upper bounds of the second-moment regularity coefficient can be expressed solely by the drift term of the linear SDE.
A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Holder continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowens arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for partially hyperbolic systems with 1-dimensional center.
In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffels theorem on irrational rotations of the circle, we compute the stack-theoretic fundamental group of hyperbolic toral automorphisms, and we revisit the classification of lens spaces.
Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $Lambda$ is center topologically mixing then $f|_{Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $ h(f)leq h(f,mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.
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