The Research of Sequence Shadowing Property and Regularly Recurrent Point on the Double Inverse Limit Space

According to the definition of sequence shadowing property and regularly recurrent point in the inverse limit space, we introduce the concept of sequence shadowing property and regularly recurrent point in the double inverse limit space and study their dynamical properties. The following results are obtained: (1) Regularly recurrent point sets of the double shift map σ f ∘ σ g are equal to the double inverse limit space of the double self-map f ∘ g in the regularly recurrent point sets. (2) The double self-map f ∘ g has sequence shadowing property if and only if the double shift map σ f ∘ σ g has sequence shadowing property. Thus, the conclusions of sequence shadowing property and regularly recurrent point are generalized to the double inverse limit space.

Segment Inequality and Almost Rigidity Structures for Integral Ricci Curvature

We will show the Cheeger–Colding segment inequality for manifolds with integral Ricci curvature bound. By using this segment inequality, the almost rigidity structure results for integral Ricci curvature will be derived by a similar method as in [1]. And the sharp Hölder continuity result of [7] holds in the limit space of manifolds with integral Ricci curvature bound.

Space of signatures as inverse limits of Carnot groups

We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in $\mathbb R^n$, as introduced by Chen. Hambly-Lyons's result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in $\mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.

Sensitivity for topologically double ergodic dynamical systems

<abstract><p>As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.</p></abstract>

The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action

First, we give the concepts of G-sequence shadowing property, G-equicontinuity and G-regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action. The following results are obtained. (1) The self-mapping f f has the G-sequence shadowing property if and only if the shift mapping σ \sigma has the G ¯ \overline{G} -sequence shadowing property; (2) The self-mapping f f is G-equicontinuous if and only if the shift mapping σ \sigma is G ¯ \overline{G} -equicontinuous; (3) R R G ¯ ( σ ) = lim ← ( R R G ( f ) , f ) R{R}_{\overline{G}}\left(\sigma )=\underleftarrow{\mathrm{lim}}\left(R{R}_{G}(f),f) . These conclusions make up for the lack of theory in the inverse limit space under group action.

On the regular-convexity of Ricci shrinker limit spaces

In this paper we study the structure of the pointed-Gromov–Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger–Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any non-collapsing Ricci shrinker limit space is strongly convex, inspired by Colding–Naber’s original idea of parabolic smoothing of the distance functions.

Ultrasonic Sensor Based Haptic Feedback Navigational System for Deaf - Blind People

Deaf-Blindness is a rare collective disorder that affects nearly 3.5 million people in today’s world. The improvement restricts the usage of two senses and by large impacts a person’s navigational capability. A variety of aid devices are used to tackle such a disability. But one common drawback that setback them is their inability to address the collective disorder. The proposed project aims at overcoming the aforementioned drawback with the help of ultrasonic sensors and haptic feedback in the form of vibrations. These sensors and feedback mechanisms are to be controlled by a microcontroller in an Arduino. Further, a rechargeable battery shall be used to accommodate the power requirements which emphasizes on energy efficiency. The project commits to limit space constraints by proposing a compact handle design and maximizes its cost efficiency such that it is affordable for everyone equally.

The structure of spaces with Bakry–Émery Ricci curvature bounded below

We explore the structure of limit spaces of sequences of Riemannian manifolds with Bakry–Émery Ricci curvature bounded below in the Gromov–Hausdorff topology. By extending the techniques established by Cheeger and Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space as in a paper of Cheeger, Colding and Tian. Our results will be applied to study the limit spaces for a sequence of Kähler metrics arising from solutions of certain complex Monge–Ampère equations for the existence problem of Kähler–Ricci solitons on a Fano manifold via the continuity method.

Lower semicontinuity of mass under C0{C^{0}} convergence and Huisken’s isoperimetric mass

Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed {C^{0}} Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken’s isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken’s work before explaining our extension of that work. The results are all specific to three dimensions.