scholarly journals Generic homeomorphisms have full metric mean dimension

2020 ◽  
pp. 1-25
Author(s):  
MARIA CARVALHO ◽  
FAGNER B. RODRIGUES ◽  
PAULO VARANDAS
Keyword(s):  
Level Set ◽  
Euclidean Distance ◽  
Dense Subset ◽  
Periodic Points ◽  
Box Dimension ◽  
Uniform Topology ◽  
Generic Subset ◽  
Interval Maps ◽  
Maximum Value ◽  
Mean Dimension

Abstract We prove that for $C^0$ -generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$ -generic homeomorphism is equal to the dimension of the manifold. In the case of continuous interval maps, we prove that each level set for the metric mean dimension with respect to the Euclidean distance is $C^0$ -dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology. Moreover, the maximum value is attained at a $C^0$ -generic subset of continuous interval maps and a dense subset of metrics topologically equivalent to the Euclidean distance.

Level Set Method Optimized with the Euclidean Distance Transform

2019 ◽  
Author(s):  
Luis Fernando Segalla ◽  
Alexandre Zabot ◽  
Diogo Nardelli Siebert ◽  
Fabiano Wolf
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Euclidean Distance ◽  
Distance Transform ◽  
Euclidean Distance Transform

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

scholarly journals Modern Dynamical Systems Theory

1974 ◽  
Vol 62 ◽  
pp. 11-18
Author(s):  
L. Markus
Keyword(s):  
Dynamical Systems ◽  
Dynamic Systems ◽  
Vector Fields ◽  
Dense Subset ◽  
Complete Metric Space ◽  
Differentiable Manifold ◽  
Open Dense Subset ◽  
Generic Properties ◽  
Countable Collection ◽  
Generic Subset

In order to analyse generic or typical properties of dynamical systems we consider the space of all C1-vector fields on a fixed differentiable manifold M. In the C1-metric, assuming M is compact, is a complete metric space and a generic subset is an open dense subset or an intersection of a countable collection of such open dense subsets of . Some generic properties (i.e. specifying generic subsets) in are described. For instance, generic dynamic systems have isolated critical points and periodic orbits each of which is hyperbolic. If M is a symplectic manifold we can introduce the space of all Hamiltonian systems and study corresponding generic properties.

scholarly journals A variational principle for the metric mean dimension of free semigroup actions

2021 ◽  
pp. 1-21
Author(s):  
MARIA CARVALHO ◽  
FAGNER B. RODRIGUES ◽  
PAULO VARANDAS
Keyword(s):  
Random Walk ◽  
Variational Principle ◽  
Metric Space ◽  
Free Semigroup ◽  
Borel Probability Measure ◽  
Box Dimension ◽  
Compact Metric Space ◽  
Semigroup Actions ◽  
Mean Dimension ◽  
The Impact

Abstract We consider continuous free semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results.

scholarly journals Some Dimensional Results of a Class of Homogeneous Moran Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jingru Zhang ◽  
Yanzhe Li ◽  
Manli Lou
Keyword(s):  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Box Dimension ◽  
Maximum Value

In this paper, we construct a class of special homogeneous Moran sets: m k -quasi-homogeneous perfect sets, and obtain the Hausdorff dimension of the sets under some conditions. We also prove that the upper box dimension and the packing dimension of the sets can get the maximum value of the homogeneous Moran sets under the condition sup k ≥ 1 m k < ∞ and estimate the upper box dimension of the sets in two cases.

scholarly journals Metric mean dimension and analog compression

2020 ◽  
Author(s):  
Yonatan Gutman ◽  
Adam Śpiewak
Keyword(s):  
Stationary Process ◽  
Rate Distortion ◽  
Upper Bounds ◽  
Regularity Conditions ◽  
Box Dimension ◽  
Lower And Upper Bounds ◽  
Distortion Functions ◽  
Stationary Stochastic Processes ◽  
Mean Dimension ◽  
Infinite Sequences

<div>Wu and Verdú developed a theory of almost lossless analog compression, where one imposes various regularity conditions on the compressor and the decompressor with the input signal being modelled by a (typically infinite-entropy) stationary stochastic process. In this work we consider all stationary stochastic processes with trajectories in a prescribed set of (bi-)infinite sequences and find uniform lower and upper bounds for certain compression rates in terms of metric mean dimension and mean box dimension. An essential tool is the recent Lindenstrauss-Tsukamoto variational principle expressing metric mean dimension in terms of rate-distortion functions. We obtain also lower bounds on compression rates for a fixed stationary process in terms of the rate-distortion dimension rates and study several examples.</div>

scholarly journals Sensitive Dependence on Initial Conditions for an Example of Markov Map: Skewed Doubling Map

MATEMATIKA ◽  
2018 ◽  
Vol 34 (1) ◽  
pp. 13-21
Author(s):  
Ummu Atiqah Mohd Roslan
Keyword(s):  
Periodic Orbits ◽  
Initial Conditions ◽  
Markov Property ◽  
Periodic Points ◽  
Numerical Results ◽  
Interval Maps ◽  
Sensitive Dependence ◽  
Markov Map ◽  
Doubling Map

Markov map is one example of interval maps where it is a piecewise ex-panding map and obeys the Markov property. One well-known example of Markov map is the doubling map, a map which has two subintervals with equal partitions. In this paper, we are interested to investigate another type of Markov map, the so-called skewed doubling map. This map is a more generalized map than the doubling map. Thus, the aims of this paper are to nd the xed points as well as the periodic points for the skewed doubling map and to investigate the sensitive dependence on initial conditions of this map. The method considered here is the cobweb diagram. Numerical results suggest that there exist dense of periodic orbits for this map. The sensitivity of this map to initial conditions is also veried where small differences in initial conditions give dierent behaviour of the orbits in the map.

scholarly journals Multifractal analysis in non-uniformly hyperbolic interval maps

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 110-133
Author(s):  
Guanzhong Ma ◽  
Wenqiang Shen ◽  
Xiao Yao
Keyword(s):  
Hausdorff Dimension ◽  
Level Set ◽  
Multifractal Analysis ◽  
Ergodic Measure ◽  
Moran Set ◽  
Interval Maps ◽  
The Given

Abstract In this paper, we establish a framework for the construction of Moran set driven by dynamics. Under this framework, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic measure in a class of non-uniformly hyperbolic interval maps with finitely many branches.

scholarly journals Simulasi dalam Optimalisasi Pengadaan Barang menggunakan Metode K-Mean Clustering

2021 ◽  
pp. 281-286
Author(s):  
Indah Savitri Hidayat ◽  
Sarjon Defit ◽  
Gunadi Widi Nurcahyo
Keyword(s):  
Data Mining ◽  
Euclidean Distance ◽  
Clustering Method ◽  
Cluster Process ◽  
Minimum Value ◽  
Maximum Value ◽  
Cluster Calculations ◽  
The Future ◽  
Clustering Data ◽  
Cluster Values

Products provided by a store have an influence on store sales. Consumers will be attracted to stores that provide products according to their wants and needs. The purpose of this research is to find out what ornamental flower products are most in demand by consumers, in demand by consumers and less desirable to consumers. Keywords: inventory of goods, K-Mean Clustering, Data Mining, cluster, optimal. Store managers can get information about goods that have been depleted of inventory stock to be updated immediately. The method used in this study is the K-Mean Clustering method which belongs to one of the branches of Data Mining. The data used in the study is data from January 2020 to December 2020 as many as 100 pieces taken from naafilah official shop, Padang. The data variables used in the entry of goods are the year, product name, price and amount sold. Furthermore, the data is processed using Rapid Miner software. The first stage of processing is to determine the value of clusters randomly, in this study researchers divided the cluster values into 3 groups. Next, the centroid value of each group will be determined. Centroid is derived from the minimum value, middle value and maximum value of the data provided. Then, the cluster process is calculated using the euclidean distance formula. Cluster calculations are done by calculating the closest distance to the data.  The final result of this study is to find out the best-selling, best-selling and less-selling ornamental flowers, so that sellers can optimize the provision of ornamental flowers for the future.

Sign in / Sign up

Export Citation Format

Share Document