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The dynamics of holomorphic maps near curves of fixed points

Filippo Bracci (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let M be a two-dimensional complex manifold and f : M → M a holomorphic map. Let S ⊂ M be a curve made of fixed points of f , i.e.  Fix ( f ) = S . We study the dynamics near  S in case  f acts as the identity on the normal bundle of the regular part of  S . Besides results of local nature, we prove that if  S is a globally and locally irreducible compact curve such that S · S < 0 then there exists a point p ∈ S and a holomorphic f -invariant curve with  p on the boundary which is attracted by  p under the action of  f . These results are achieved...

The dynamics of two-circle and three-circle inversion

Daniel M. Look (2008)

Fundamenta Mathematicae

We study the dynamics of a map generated via geometric circle inversion. In particular, we define multiple circle inversion and investigate the dynamics of such maps and their corresponding Julia sets.

The entropy conjecture for diffeomorphisms away from tangencies

Gang Liao, Marcelo Viana, Jiagang Yang (2013)

Journal of the European Mathematical Society

We prove that every C 1 diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.

The entropy of algebraic actions of countable torsion-free abelian groups

Richard Miles (2008)

Fundamenta Mathematicae

This paper is concerned with the entropy of an action of a countable torsion-free abelian group G by continuous automorphisms of a compact abelian group X. A formula is obtained that expresses the entropy in terms of the Mahler measure of a greatest common divisor, complementing earlier work by Einsiedler, Lind, Schmidt and Ward. This leads to a uniform method for calculating entropy whenever G is free. In cases where these methods do not apply, a possible entropy formula is conjectured. The entropy...

The equivalence of controlled lagrangian and controlled hamiltonian systems

Dong Eui Chang, Anthony M. Bloch, Naomi E. Leonard, Jerrold E. Marsden, Craig A. Woolsey (2002)

ESAIM: Control, Optimisation and Calculus of Variations

The purpose of this paper is to show that the method of controlled lagrangians and its hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity)...

The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems

Dong Eui Chang, Anthony M. Bloch, Naomi E. Leonard, Jerrold E. Marsden, Craig A. Woolsey (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity)...

The existence of limit cycle for perturbed bilinear systems

Hanen Damak, Mohamed Ali Hammami, Yeong-Jeu Sun (2012)

Kybernetika

In this paper, the feedback control for a class of bilinear control systems with a small parameter is proposed to guarantee the existence of limit cycle. We use the perturbation method of seeking in approximate solution as a finite Taylor expansion of the exact solution. This perturbation method is to exploit the “smallness” of the perturbation parameter ε to construct an approximate periodic solution. Furthermore, some simulation results are given to illustrate the existence of a limit cycle for...

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