arXiv subjects
Publications and source records attributed to Jeovanny de Jesus Muentes Acevedo.
We study Anosov families which are sequences of diffeomorphisms along compact Riemannian manifolds such that the tangent bundle split into expanding and contracting subspaces. In this paper we prove that a certain class of Anosov families: (i) admit canonical coordinates (ii) are expansive, (iii) satisfy the shadowing property, and (iv) exhibit a Markov partition.
It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities. Let $N$ be an $n$-dimensional compact Riemannian manifold, where $n \geq 2$, and $α\in [0, n]$. In this paper, we construct a homeomorphism $ϕ: N \rightarrow N$ with mean Hausdorff dimension equal to $α$. Furthermore, we prove that the set of homeomorphisms on $N$ with both lower and upper mean Hausdorff dimensions equal to $α$ is dense in $\text{Hom}(N)$. Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to $n$ contains a residual subset of $\text{Hom}(N).$
Let $N$ be an $n$-dimensional compact riemannian manifold, with $n\geq 2$. In this paper, we prove that for any $α\in [0,n]$, the set consisting of homeomorphisms on $N$ with lower and upper metric mean dimensions equal to $α$ is dense in $\text{Hom}(N)$. More generally, given $α,β\in [0,n]$, with $α\leq β$, we show the set consisting of homeomorphisms on $N$ with lower metric mean dimension equal to $α$ and upper metric mean dimension equal to $β$ is dense in $\text{Hom}(N)$. Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to $n$ is residual in $\text{Hom}(N)$.
M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if $N$ is a $n$ dimensional compact riemannian manifold then, for any $a\in [0,n]$, the set consisting of continuous maps with metric mean dimension equal to $a$ is dense in $C^{0}(N)$ and for $a=n$ this set is residual. Furthermore, we prove some results related to existence and density of continuous maps on Cantor sets with positive metric mean dimension and on product spaces with positive mean dimension.
In this paper we extend the definitions of mean dimension and metric mean di-mension for non-autonomous dynamical systems. We show some properties of this extension and furthermore some applications to the mean dimension and metric mean dimension of single continuous maps.
Anosov families are non-stationary dynamical systems with hyperbolic behavior. Non-trivial examples of Anosov families will be given in this paper. We show the existence of invariant manifolds, the structrural stability and a characterization for a certain class of Anosov families.
In the late 1990's, M. Gromov introduced the notion of mean dimension for a continuous map, which is, as well as the topological entropy, an invariant under topological conjugacy. The concept of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000. In this paper we will verify which properties coming from the topological entropy map are valid for both mean dimension and metric mean dimension. In particular, we will prove that the metric mean dimension map is not continuous anywhere on the set consisting of continuous maps on both the Cantor set, the interval or the circle. Finally we prove that the metric mean dimension on the set consisting of continuous map on the interval and on the circle is not lower semi-continuous.
Simple Lie algebras of finite dimension over an algebraically closed field of characteristic 0 or $p> 3$ were recently classified. However, the problem over an algebraically closed field of characteristics 2 or 3 there exist only partial results. The first result on the problem of classification of simple Lie algebra of finite dimension over an algebraically closed field of characteristic 2 is that these algebras have absolute toral rank greater than or equal to 2. In this paper we show that there are not simple Lie 2-algebras with toral rank 3 over an algebraically closed field of characteristic 2 and dimension less or equal to 16.
Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. Roughly, an Anosov family is a two-sided sequence of diffeomorphisms (or non-stationary dynamical system) with similar behavior to an Anosov diffeomorphisms. We show that the set consisting of Anosov families is an open subset of the set consisting of two-sided sequences of diffeomorphisms, which is endowed with the strong topology (or Whitney topology).
Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. In addition to presenting several properties and examples of Anosov families, in this paper we build local stable and local manifolds for such families.
Let $M$ be a compact Riemannian manifold. The set $\text{F}^{r}(M)$ consisting of sequences $(f_{i})_{i\in\mathbb{Z}}$ of $C^{r}$-diffeomorphisms on $M$ can be endowed with the compact topology or with the strong topology. A notion of topological entropy is given for these sequences. I will prove this entropy is discontinuous at each sequence if we consider the compact topology on $\text{F}^{r}(M)$. On the other hand, if $ r\geq 1$ and we consider the strong topology on $\text{F}^{r}(M)$, this entropy is a continuous map.
Let H be a real (or complex) Hilbert space. Every nonnegative operator $L \in L(H)$ admits a unique nonnegative square root $R \in L(H)$, i.e., a nonnegative operator $R \in L(H)$ such that $R^{2}= L$. Let $GL^{+}_{S}(H)$ be the set of nonnegative isomorphisms in $L(H)$. First we will show that $GL^{+}_{S}(H)$ is a convex (real) Banach manifold. Denoting by $L^{1/2}$ the nonnegative square root of $L$. In [10], Richard Bouldin proves that $L^{1/2}$ depends continuously on $L$ (this proof is non-trivial). This result has several applications. For example, it is used to find the polar decomposition of a bounded operator. This polar decomposition allows us to determine the positive and negative spectral subespaces of any self-adjoint operator, and moreover, allows us to define the Maslov index. The autor of the paper under review provides an alternative proof (and a little more simplified) that $L^{1/2}$ depends continuously on $L$, and moreover, he shows that the map \begin{align}R &: GL^{+}_{S}(H)\rightarrow GL^{+}_{S}(H)\\ L &\to L^{1/2} \end{align} is a homeomorphism.