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We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.

We say that a group G of local (maybe formal) biholomorphisms satisfies the uniform intersection property if the intersection multiplicity $(\\phi (V), W)$ takes only finitely many values as a function of G for any choice of analytic sets V and W of complementary dimension. In dimension $2$ we show that G satisfies the uniform intersection property if and only if it is finitely determined – that is, if there exists a natural number k such that different elements of G have different Taylor expansions of degree k at the origin. We also prove that G is finitely determined if and only if the action of G on the space of germs of analytic curves has discrete orbits.

In this paper we study a non-cooperative sequential equilibrium concept, namely the Stackelberg–Nash equilibrium, in a game in which heterogeneous atomic traders interact in interrelated markets. To this end, we consider a two-stage quantity setting strategic market game with a finite number of traders. Within this framework, we define a Stackelberg–Nash equilibrium. Then, we show existence and local uniqueness of a Stackelberg–Nash equilibrium with trade. To this end, we use a differentiable approach: the vector mapping which determines the strategies of followers is a smooth local diffeomorphism, and the set of Stackelberg–Nash equilibria with trade is discrete, i.e., the interior equilibria of the game are locally unique. We also compare through examples the sequential and the simultaneous moves games. A striking difference is that exchange can take place in one subgame while autarky can hold in another subgame, in which case only leaders (followers) make trade.

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric inn+2dimensions that encodes a conformal class of metrics inndimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric inn+1dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

The paper presents a transformation of nonlinear electric circuit into linear one through changing coordinates (local diffeomorphism) with the use of closed feedback loop. The necessary conditions that must be fulfilled by nonlinear system to enable carrying out linearizing procedures are presented. Numerical solutions of state equations for the nonlinear system and equivalent linearized system are included.

In 1920, Pierre Fatou expressed the conjecture that--except for special cases--all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. This conjecture remains the main open problem in the dynamics of iterated maps. For the logistic family x- ax(1-x), it can be interpreted to mean that for a dense set of parameters \"a,\" an attracting periodic orbit exists. The same question appears naturally in science, where the logistic family is used to construct models in physics, ecology, and economics. In this book, Jacek Graczyk and Grzegorz Swiatek provide a rigorous proof of the Real Fatou Conjecture. In spite of the apparently elementary nature of the problem, its solution requires advanced tools of complex analysis. The authors have written a self-contained and complete version of the argument, accessible to someone with no knowledge of complex dynamics and only basic familiarity with interval maps. The book will thus be useful to specialists in real dynamics as well as to graduate students.

Necessary and sufficient conditions are presented under which a discrete-time autonomous system with outputs is locally diffeomorphic to an output-scaled linear observable system or an output-scaled nonlinear system in the observer form. As a consequence of such characterizations, the nonlinear observer design problem is studied for a broader class of discrete-time nonlinear systems by using the exact linearization technique that is based on the differential geometric approach.

In 1920, Pierre Fatou expressed the conjecture that--except for special cases--all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. This conjecture remains the main open problem in the dynamics of iterated maps. For the logistic family x- ax(1-x), it can be interpreted to mean that for a dense set of parameters \"a,\" an attracting periodic orbit exists. The same question appears naturally in science, where the logistic family is used to construct models in physics, ecology, and economics. In this book, Jacek Graczyk and Grzegorz Swiatek provide a rigorous proof of the Real Fatou Conjecture. In spite of the apparently elementary nature of the problem, its solution requires advanced tools of complex analysis. The authors have written a self-contained and complete version of the argument, accessible to someone with no knowledge of complex dynamics and only basic familiarity with interval maps. The book will thus be useful to specialists in real dynamics as well as to graduate students.

In this paper, we develop a machine which enables us to predict, in many cases, the exact number of fixed points of a local diffeomorphism. Though much more general, our technique applies in particular to locally expansive maps on compact, connected, orientable differentiable manifolds.

It is well known that a theory with explicit Lorentz violation is not invariant under diffeomorphisms. On the other hand, for geometrical theories of gravity, there are alternative transformations, which can be best defined within the first-order formalism and that can be regarded as a set of improved diffeomorphisms. These symmetries are known as local translations, and among other features, they are Lorentz covariant off shell. It is thus interesting to study if theories with explicit Lorentz violation are invariant under local translations. In this work, an example of such a theory, known as the minimal gravity sector of the Standard Model Extension, is analyzed. Using a robust algorithm, it is shown that local translations are not a symmetry of the theory. It remains to be seen if local translations are spontaneously broken under spontaneous Lorentz violation, which are regarded as a more natural alternative when spacetime is dynamic.