INVERSE TOPOLOGICAL PRESSURE WITH APPLICATIONS TO HOLOMORPHIC DYNAMICS OF SEVERAL COMPLEX VARIABLES

2004 ◽  
Vol 06 (04) ◽  
pp. 653-679 ◽  
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Technical Condition ◽  
Topological Pressure ◽  
Holomorphic Dynamics ◽  
Critical Set ◽  
Axiom A ◽  
Type Definition ◽  
Basic Set ◽  
Unique Zero ◽  
Stable Dimension ◽  
The One

In this paper, we introduce a few notions of inverse topological pressure [Formula: see text], defined in terms of backward orbits (prehistories) instead of forward orbits. This inverse topological pressure has some properties similar to the regular (forward) pressure but, in general, if the map is not a homeomorphism, they do not coincide. In fact, there are several ways to define inverse topological pressure; for instance, we show that the Bowen type definition coincides with the one using spanning sets. Then we consider the case of a holomorphic map [Formula: see text] which is Axiom A and such that its critical set does not intersect a particular basic set of saddle type Λ. We will prove that, under a technical condition, the Hausdorff dimension of the intersection between the local stable manifold and the basic set is equal to ts, i.e. [Formula: see text], for all points x belonging to Λ. Here ts represents the unique zero of the function t→P-(tϕs), with P- denoting the inverse topological pressure and [Formula: see text], y∈Λ. In general, [Formula: see text] will be estimated above by ts and below by [Formula: see text], where [Formula: see text] is the unique zero of the map t→P_(tϕs). As a corollary we obtain that, if the stable dimension is non-zero, then Λ must be a non-Jordan curve, and also, if f|Λ happens to be a homeomorphism (like in the examples from [13]), then the stable dimension cannot be zero.

scholarly journals Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps

2008 ◽  
Vol 60 (3) ◽  
pp. 658-684 ◽  
Author(s):  
Eugen Mihailescu ◽  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Arbitrary Point ◽  
Base Point ◽  
Topological Pressure ◽  
Axiom A ◽  
Hyperbolic Diffeomorphisms ◽  
Base Points ◽  
Basic Set ◽  
Fixed Positive Number ◽  
Stable Dimension

AbstractWe study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) mapf: ℙ2ℂ → ℙ2ℂ, where ℙ2ℂ stands for the complex projective space of dimension 2. Letδs(x)denote a basic set for f of unstable index 1, and x an arbitrary point of Λ; we denote byδs(x)the Hausdorff dimension of∩ Λ, whereris some fixed positive number andis the local stable manifold atxof sizer;δs(x)is calledthe stable dimension at x. Mihailescu and Urba ńnski introduced a notion of inverse topological pressure, denoted by P−, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates off, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on Λ. When each pointxfrom Λ has the same numberd′of preimages in Λ, then we show thatδs(x)is independent of x; in factδs(x)is shown to be equal in this case with the unique zero of the mapt → P(tϕs−log d′). We also prove the Lipschitz continuity of the stable vector spaces over Λ; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

scholarly journals Homology for one-dimensional solenoids

2017 ◽  
Vol 121 (2) ◽  
pp. 219 ◽  
Author(s):  
Massoud Amini ◽  
Ian F. Putnam ◽  
Sarah Saeidi Gholikandi
Keyword(s):  
Finite Type ◽  
Homology Theory ◽  
One Dimensional ◽  
Axiom A ◽  
Group Invariant ◽  
Dimension Group ◽  
Basic Set ◽  
The One ◽  
Topological Dynamical Systems ◽  
Smale Spaces

Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Arcabouço de Comunicação Publish/Subscribe para a Coleta de Dados de Aplicações Médicas

2021 ◽  
Author(s):  
Gabriele Ribeiro ◽  
Joberto S. B. Martins
Keyword(s):  
Network Structure ◽  
Medical Data ◽  
Medical Applications ◽  
Distributed Environment ◽  
Communication Framework ◽  
Production And Consumption ◽  
Basic Set ◽  
Network Communications ◽  
The One ◽  
Health Application

Medical applications are increasingly using computing resources such as IoT sensors and network communications paradigms. An e-Health application requires a basic set of elements such as sensors, a communication framework, and a network structure adapted to the application's specific requirements. This work expands and develops a framework based on the Publish / Subscribe paradigm to develop PSIoT-Health. The PSIoT-Health framework focuses on medical applications that collect data produced in a distributed manner. The PSIoT-Health adapts the Pub/Sub model to the requirements of medical applications and proposes a solution for the production and consumption of data between producers and consumers of medical data in a distributed environment such as the one existing in a smart city.

Metric properties of some fractal sets and applications of inverse pressure

2009 ◽  
Vol 148 (3) ◽  
pp. 553-572 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Hausdorff Dimension ◽  
General Setting ◽  
Topological Pressure ◽  
Box Dimension ◽  
Stable Manifolds ◽  
Fractal Sets ◽  
Metric Properties ◽  
Basic Set ◽  
Unstable Set ◽  
Unstable Manifolds

AbstractWe consider iterations of smooth non-invertible maps on manifolds of real dimension 4, which are hyperbolic, conformal on stable manifolds and finite-to-one on basic sets. The dynamics of non-invertible maps can be very different than the one of diffeomorphisms, as was shown for example in [4,7,12,17,19], etc. In [13] we introduced a notion of inverse topological pressureP−which can be used for estimates of the stable dimension δs(x) (i.e the Hausdorff dimension of the intersection between the local stable manifoldWsr(x) and the basic set Λ,x∈ Λ). In [10] it is shown that the usual Bowen equation is not always true in the case of non-invertible maps. By using the notion of inverse pressureP−, we showed in [13] that δs(x) ≤ts(ϵ), wherets(ϵ) is the unique zero of the functiont→P−(tφs, ϵ), for φs(y):= log|Dfs(y)|,y∈ Λ and ϵ > 0 small. In this paper we prove that if Λ is not a repellor, thents(ϵ) < 2 for any ϵ > 0 small enough. In [11] we showed that a holomorphic s-hyperbolic map on2has a global unstable set with empty interior. Here we show in a more general setting than in [11], that the Hausdorff dimension of the global unstable setWu() is strictly less than 4 under some technical derivative condition. In the non-invertible case we may have (infinitely) many unstable manifolds going through a point in Λ, and the number of preimages belonging to Λ may vary. In [17], Qian and Zhang studied the case of attractors for non-invertible maps and gave a condition for a basic set to be an attractor in terms of the pressure of the unstable potential. In our case the situation is different, since the local unstable manifolds may intersect both inside and outside Λ and they do not form a foliation like the stable manifolds. We prove here that the upper box dimension ofWsr(x) ∩ Λ is less thants(ϵ) for any pointx∈ Λ. We give then an estimate of the Hausdorff dimension ofWu() by a different technique, using the Holder continuity of the unstable manifolds with respect to their prehistories.

Deutsche Bahn AG: a former monopoly off track?

2017 ◽  
Vol 13 (1) ◽  
pp. 25-58 ◽  
Author(s):  
Carolin Berlich ◽  
Felix Daut ◽  
Anna C. Freund ◽  
Andrea Kampmann ◽  
Benedict Killing ◽  
...  
Keyword(s):  
Strategic Management ◽  
Technical Condition ◽  
Basic Knowledge ◽  
Long Distance ◽  
Public Companies ◽  
Content Type ◽  
Customer Complaints ◽  
Political Events ◽  
Deutsche Bahn ◽  
The One

Synopsis Deutsche Bahn AG (Deutsche Bahn hereafter) was the former German railroad monopolist until deregulation in 1996. It was a well-known company that operated in worldwide markets for transport and logistics at the time of the case (late 2013). The case “Deutsche Bahn AG: a former monopoly off track?” focuses on the opportunities and challenges faced by Deutsche Bahn with regard to its position in the German individual transportation market. On the one hand, Deutsche Bahn is facing external problems. Increasing competition in short- and long-distance traffic threatens its strong business position. The competition emerged from a growing long-distance bus market and the increase in private railway companies. During the last few years before 2013, Deutsche Bahn has lost several public tenders for individual passenger travel in Germany. On the other hand, Deutsche Bahn has internal problems that endanger its image as a service company. A lack of service quality and the technical condition of its trains has led to rising numbers of customer complaints. In addition, staffing and punctuality problems have exacerbated the situation. One of the main technical issues the company faces is that ordered trains have not been delivered on time. Given the focus on Deutsche Bahn’s domestic challenges, its international business activities are tackled only briefly. While regulatory and political events have an impact on Deutsche Bahn, these are not the main subjects of the case. Research methodology This case has been written from public sources. Consequently, no company release is provided. None of the information has been disguised in any way. Relevant courses and levels The case is intended for use in a 90-minute strategic management class attended by students at the end of their undergraduate studies or in postgraduate study. Although the case relates to issues in strategic management, the special regulatory environment and some of the issues covered could make the case a useful complement in other classes as well, such as classes in supply chain management (procurement) or the management of public companies. Therefore, students should have basic knowledge in developing strategies, management, marketing, human resource management, and finance. Theoretical bases Strategic Analysis and Strategic Management, Railroad Logistics, Deregulation of a former Monopoly, Stakeholder Theory.

A framework toward understanding the characterization of holomorphic dynamics

2014 ◽  
Author(s):  
Yunping Jiang
Keyword(s):  
Hyperbolic Geometry ◽  
Spherical Geometry ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Holomorphic Maps ◽  
Critical Set ◽  
Geometrically Finite ◽  
Infinite Degree ◽  
New Framework

This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of hyperbolic distortion in hyperbolic geometry, the framework suggested here is based on controlling conformal distortion in spherical geometry. The new framework enables one to relax two of Thurston's assumptions: first, that the iterated map has finite degree and, second, that its post-critical set is finite. Thus, it makes possible to characterize certain rational maps for which the post-critical set is not finite as well as certain classes of entire and meromorphic coverings for which the iterated map has infinite degree.

Mathematical model of the clutch of the Volkswagen Polo Sedan as an object of diagnosis

2021 ◽  
Vol 13 (1) ◽  
pp. 23-32
Author(s):  
Dmytro Borysiuk ◽  
Keyword(s):  
Mathematical Model ◽  
Block Diagram ◽  
Technical Condition ◽  
Mechanical Transmission ◽  
Traffic Lights ◽  
Operating Rules ◽  
External Feature ◽  
The One ◽  
Urban Conditions ◽  
Strength And Durability

When designing a car, designers must find a compromise between the dynamism and economy of the car, on the one hand, and reliability and safety - on the other. The first problem is solved mainly by reducing the weight of the car by optimizing the design and reducing strength reserves, and the second - by increasing the strength and durability of structural elements. Reliability of units of cars is the maximum at work on steady modes that confirms experience of their operation. It is known that in the conditions of country routes on good roads reliability of cars is essentially higher, than at operation within the city. In cities, unstable modes of operation of car units are caused, first of all, by frequent stops and subsequent accelerations due to the presence of traffic lights, pedestrian crossings, railway crossings, traffic jams, steep ascents and descents on car routes, etc. All this necessitates frequent braking, shifting gears and moving. These circumstances lead to an increase in the dynamic load of the vehicle units and, as a consequence, to the intensification of the processes of wear of the friction pairs of the clutch discs, gears and cardan gears, and so on. All the factors that reduce the reliability of car units in urban conditions are particularly evident in the example of buses and taxis with a manual transmission. According to some data, the share of failures of mechanical transmission units of such cars is 19 ... 23% of all failures. During intensive operation of the car, various clutch malfunctions can occur. There are malfunctions of the actual clutch and malfunction of the clutch drive. Coupling defects occur not only due to intensive operation, but also due to violations of operating rules. Currently, clutch failures are mainly diagnosed by external signs. However, one external feature may correspond to several clutch failures. Thus, the process of determining the technical condition of the clutch of vehicles is an urgent scientific and technical task. A mathematical model of the clutch of the Volkswagen Polo Sedan as an object of diagnosis are presents in the article. A matrix for diagnosing the clutch of a Volkswagen Polo Sedan and a block diagram of its synthesis have been built.

The ambient structure of basic sets

1995 ◽  
Vol 15 (6) ◽  
pp. 1183-1188
Author(s):  
A. Löffler
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Product Structure ◽  
Axiom A ◽  
Basic Set ◽  
Local Product ◽  
Basic Sets

AbstractLet Λ be a basic set of an Axiom A diffeomorphism of a compact Riemannian manifold M without boundary. If ε is small enough one can find by local product structure that for x ε Λ there is a neighborhood V(x) in M such that V ∩ Λ is homeomorphic to . The author proves that this homeomorphism can be extended to a homeomorphism of V onto .

scholarly journals Effectiveness of transposed inverse sets in faber regions

1983 ◽  
Vol 6 (2) ◽  
pp. 285-295 ◽  
Author(s):  
J. A. Adepoju ◽  
M. Nassif
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Main Result Theorem ◽  
Simple Set ◽  
Other Hand ◽  
Basic Set ◽  
Result Theorem ◽  
The One ◽  
The Given ◽  
Class Of Functions

The effectiveness properties, in Faber regions, of the transposed inverse of a given basic set of polynominals, are investigated in the present paper. A certain inevitable normalizing substitution, is first formulated, to be undergone by the given set to ensure the existence of the transposed inverse in the Faber region. The first main result of the present work (Theorem 2.1), on the one hand, provides a lower bound of the class of functions for which the normalized transposed inverse set is effective in the Faber region. On the other hand, the second main result (Theorem 5.2) asserts the fact that the normalized transposed inverse set of a simple set of polynomials, which is effective in a Faber region, should not necessarily be effective there.

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