Types d'orbites et dynamique minimale pour les applicationes continues de graphes

We define the type of a periodic orbit of a graph map. We consider the class of ‘train-track’ representatives, that is, those graph maps which minimize the topological entropy of the topological representatives of a given free group endomorphism. We prove that each type of periodic orbit realized by an efficient representative is also realised by any representative of the same free group endomorphism. Moreover, the number of periodic orbits of a given type is minimized by the efficient representatives

On the topological dynamics and phase-locking renormalization of Lorenz-Like maps

The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz--like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz-- like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase--locking renormalizable maps. This characterization is given by showing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an $*$--like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map

On the topological dynamics and phase-locking renormalization of Lorenz-Like maps

The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz--like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz-- like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase--locking renormalizable maps. This characterization is given by showing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an $*$--like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map

On the topological dynamics and phase-locking renormalization of Lorenz-Like maps

The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz--like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz-- like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase--locking renormalizable maps. This characterization is given by showing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an $*$--like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map

On the primary orbits of star maps

This paper is the first one of a series of two, in which we characterizea class of primary orbits of self maps of the 4-star with the branching pointfixed. This class of orbits plays, for such maps, the same role as the directedprimary orbits of self maps of the 3-star with the branching point fixed. Someof the primary orbits (namely, those having at most one coloured arrow) arecharacterized at once for the general case of n-star maps.This paper is the second part of [1] and is devoted to the study of thespiral orbits of self maps of the 4-star with the branching point fixed, completingthe characterization of the strongly directed primary orbits for such map

On the primary orbits of star maps

This paper is the first one of a series of two, in which we characterizea class of primary orbits of self maps of the 4-star with the branching pointfixed. This class of orbits plays, for such maps, the same role as the directedprimary orbits of self maps of the 3-star with the branching point fixed. Someof the primary orbits (namely, those having at most one coloured arrow) arecharacterized at once for the general case of n-star maps.This paper is the second part of [1] and is devoted to the study of thespiral orbits of self maps of the 4-star with the branching point fixed, completingthe characterization of the strongly directed primary orbits for such map

Rotation sets for orbits of degree one circle maps

Let F be the lifting of a circle map of degree one. In [R. Bam´on, I. P. Malta, M. J. Pacifico and F.Takens, Ergodic Theory Dyn. Syst. 4, 493-498 (1984; Zbl 0605.58027)], a notion of F-rotation interval of a point x 2 S1 was given. In this paper, we define and study a new notion of a rotation set of a point which preserves more of the dynamical information contained in the sequences {Fn(y)}1n=0 than the one preserved from [R. Bam´on et al., loc. cit.]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the main theorem in [Bam´on et al., 1984] in our settings

Rotation sets for orbits of degree one circle maps

Let F be the lifting of a circle map of degree one. In [R. Bam´on, I. P. Malta, M. J. Pacifico and F.Takens, Ergodic Theory Dyn. Syst. 4, 493-498 (1984; Zbl 0605.58027)], a notion of F-rotation interval of a point x 2 S1 was given. In this paper, we define and study a new notion of a rotation set of a point which preserves more of the dynamical information contained in the sequences {Fn(y)}1n=0 than the one preserved from [R. Bam´on et al., loc. cit.]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the main theorem in [Bam´on et al., 1984] in our settings