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Dynamics on Hubbard trees

100%
Alsedà L., Fagella N.
Fundamenta Mathematicae
|
2000
|
tom 164
|
nr 2
115-141
EN
It is well known that the Hubbard tree of a postcritically finite complex polynomial contains all the combinatorial information on the polynomial. In fact, an abstract Hubbard tree as defined in [23] uniquely determines the polynomial up to affine conjugation. In this paper we give necessary and sufficient conditions enabling one to deduce directly from the restriction of a quadratic Misiurewicz polynomial to its Hubbard tree whether the polynomial is renormalizable, and in this case, of which type. Moreover, we study dynamical features such as entropy, transitivity or periodic structure of the polynomial restricted to the Hubbard tree, and compare them with the properties of the polynomial on its Julia set. In other words, we want to study how much of the "dynamical information" about the polynomial is captured by the Hubbard tree.
2
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On the primary orbits of star maps (second part: spiral orbits)

100%
Alsedà L., Moreno J.
Applicationes Mathematicae
|
2002
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tom 29
|
nr 3
331-357
EN
This paper is the second part of [2] and is devoted to the study of the spiral orbits of self maps of the 4-star with the branching point fixed, completing the characterization of the strongly directed primary orbits for such maps.
3
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On the definition of strange nonchaotic attractor

100%
Alsedà L., Costa S.
Fundamenta Mathematicae
|
2009
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tom 206
|
nr 1
23-39
EN
The aim of this paper is twofold. On the one hand, we want to discuss some methodological issues related to the notion of strange nonchaotic attractor. On the other hand, we want to formulate a precise definition of this kind of attractor, which is "observable" in the physical sense and, in the two-dimensional setting, includes the well known models proposed by Grebogi et al. and by Keller, and a wide range of other examples proposed in the literature. Furthermore, we analytically prove that a whole family of two-dimensional quasiperiodic skew products defined on 𝕊¹ × ℝ have strange nonchaotic attractors. As a corollary we show analytically that the system proposed by Grebogi et al. has a strange nonchaotic attractor.
4
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All solenoids of piecewise smooth maps are period doubling

81%
Alsedà L., López V. J., L'ubomír Snoha
Fundamenta Mathematicae
|
1998
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tom 157
|
nr 2-3
121-138
EN
We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if $p_1 < ... < p_n$ is a periodic orbit of a continuous map f then there is a union set ${q_1,..., q_{n-1}}$ of some periodic orbits of f such that $p_i < q_i < p_{i+1}$ for any i.
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