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Kneading theory of Lorenz maps

100%
Alsedà L., Llibre J.
Banach Center Publications
|
1989
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tom 23
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nr 1
83-89
2
Content available remote

Results and open questions on some invariants measuring the dynamical complexity of a map

100%
Llibre J., Saghin R.
Fundamenta Mathematicae
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2009
|
tom 206
|
nr 1
307-327
EN
Let f be a continuous map on a compact connected Riemannian manifold M. There are several ways to measure the dynamical complexity of f and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of f, the volume growth of f, the rate of growth of periodic points of f, some invariants related to exterior powers of the derivative of f, and several invariants measuring the topological complexity of f: the degree (for the case when the manifold is orientable), the spectral radius of the map induced by f on the homology of M, the fundamental-group entropy, the asymptotic Lefschetz number and the asymptotic Nielsen number. In general these relations depend on the smoothness of f. Various examples are provided.
3
Content available remote

Periods of Morse-Smale diffeomorphisms of 𝕊²

100%
Guirao J., Llibre J.
Colloquium Mathematicum
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2008
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tom 110
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nr 2
477-483
EN
The aim of this paper is to describe the set of periods of a Morse-Smale diffeomorphism of the two-dimensional sphere according to its homotopy class. The main tool for proving this is the Lefschetz fixed point theory.
4
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Dynamics of a Lotka-Volterra map

81%
Balibrea F., Guirao J., Lampart M., Llibre J.
Fundamenta Mathematicae
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2006
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tom 191
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nr 3
265-279
EN
Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F: (x,y) ↦ (x(4-x-y),xy) introduced by A. N. Sharkovskiĭ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior of the Schrödinger equation.
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