Homoclinic Points of Principal Algebraic Actions

2016 ◽  
pp. 251-292
Author(s):  
Martin Göll ◽  
Evgeny Verbitskiy
Keyword(s):  
Homoclinic Points ◽  
Algebraic Actions

scholarly journals Algebraic actions of the discrete Heisenberg group: Expansiveness and homoclinic points

2014 ◽  
Vol 25 (4) ◽  
pp. 713-744 ◽  
Author(s):  
Martin Göll ◽  
Klaus Schmidt ◽  
Evgeny Verbitskiy
Keyword(s):  
Heisenberg Group ◽  
Homoclinic Points ◽  
Algebraic Actions ◽  
Discrete Heisenberg Group

scholarly journals Homoclinic points, atoral polynomials, and periodic points of algebraic -actions

2012 ◽  
Vol 33 (4) ◽  
pp. 1060-1081 ◽  
Author(s):  
DOUGLAS LIND ◽  
KLAUS SCHMIDT ◽  
EVGENY VERBITSKIY
Keyword(s):  
Growth Rate ◽  
Main Tool ◽  
Periodic Points ◽  
Laurent Polynomials ◽  
Homoclinic Points ◽  
Complex Variety ◽  
Finite Set ◽  
Algebraic Actions ◽  
The Ideal

AbstractCyclic algebraic ${\mathbb {Z}^{d}}$-actions are defined by ideals of Laurent polynomials in $d$ commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative $d$-torus. For such expansive actions it is known that the limit of the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the $d$-torus in a finite set. Here we further extend it to the case where the dimension of intersection of the variety with the $d$-torus is at most $d-2$. The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

scholarly journals New examples of Bernoulli algebraic actions

2021 ◽  
pp. 1-12
Author(s):  
DOUGLAS LIND ◽  
KLAUS SCHMIDT
Keyword(s):  
Abelian Group ◽  
Large Class ◽  
Free Group ◽  
Compact Abelian Group ◽  
Homoclinic Points ◽  
Independent Generator ◽  
Algebraic Action ◽  
Algebraic Actions ◽  
Shift Action

Abstract We give an example of a principal algebraic action of the non-commutative free group ${\mathbb {F}}$ of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of ${\mathbb {F}}$ . The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of ${\mathbb {F}}$ without an obvious independent generator. Our methods can be generalized to a large class of acting groups.

scholarly journals On the creation of homoclinic points

1987 ◽  
Vol 66 (1) ◽  
pp. 139-159 ◽  
Author(s):  
Ricardo Mañé
Keyword(s):  
Homoclinic Points ◽  
The Creation

scholarly journals Mine surveying technique and steel storage tank base deformation removal

2021 ◽  
Vol 1 (7) ◽  
pp. 69-79
Author(s):  
Aleksandr I. Barulin ◽  
Keyword(s):  
Low Cost ◽  
Operating Time ◽  
Construction Period ◽  
Steel Tank ◽  
Steel Tanks ◽  
Algebraic Actions ◽  
Minimum Height ◽  
Survey Accuracy ◽  
Tank Bottom ◽  
Mine Surveying

Introduction. Vertical steel tanks are extensively used in oil fields to hold crude oil and other liquids. Their construction requires continuous surveying control. However, measurement procedures, data processing, and deviations and documentation correction are not universal and cannot be applied for tanks with a capacity of less than 3000 m3. Moreover, the process of correcting the detected bottom irregularities is poorly substantiated and intuitive. Research objective is to improve the as-built survey accuracy and reliability of the low-capacity steel tank bottom, substantiate the minimum height of its irregularities, increase the objectivity and productivity of measurement processing, develop an unambiguous method for time-predictable correction of bottom irregularities, and substantiate the optimal contents of its relief’s as-built scheme. Methods of research. Geometric leveling for the bottom profile survey failed to provide adequate accuracy and was therefore replaced by the method of trigonometric leveling. A method has been developed for determining the deviations of the existent tank bottom profile from the design position by means of algebraic actions with surfaces of a topographic order. The accuracy of determining the smallest height of bottom irregularities has been estimated under the tacheometric survey. A method has been proposed for controlling the correction of tank bottom irregularities based on evaluation calculations of bottom deformations by the finite element method. Research results. All the development results are exemplified by a specific production example. It was found that for a full completion of work, two iterations of tank bottom irregularities correction are enough. Methods of optimal design for the facility’s as-built schemes are presented. Conclusions. A simple, accurate, low-cost, productive, and time-predictable method of mine surveying, mathematical processing and correction of deviations in the tank bottom profile has been developed. This technique reduces the construction period and increases the operating time of the facility.

Homoclinic Points and Bifurcations

2021 ◽  
pp. 135-142
Author(s):  
Robert L. Devaney
Keyword(s):  
Homoclinic Points
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