We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict ...traditional sense, as invariant compact trees embedded in
$\mathbb {C}$
, do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a homotopy Hubbard tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a homotopy Hubbard tree that is unique up to homotopy, and that post-singularly finite exponential maps can be classified in terms of homotopy Hubbard trees, using a transcendental analogue of Thurston’s topological characterization theorem of rational maps.
The strange star product Alvin, Lori
Journal of difference equations and applications,
04/2012, Letnik:
18, Številka:
4
Journal Article
Recenzirano
In this paper, we introduce the strange star product. This product may be used to locate non-infinitely renormalizable unimodal maps f where
is topologically conjugate to an adding machine. Given any ...sequence
, with each
, the strange star product can precisely construct the kneading sequence of a unimodal map with an embedded copy of the associated α-adic adding machine.
A strange adding machine is a non-renormalizable unimodal map,
f, with critical point
c, such that
f
|
ω
(
c
)
is topologically conjugate to an adding machine map. In this paper we characterize the ...kneading sequence structure for all strange adding machines.
This paper is contributed to the combinatorial properties of the periodic kneading words of antisymmetric cubic maps defined on a interval. The least words of given lengths, the adjacency relations ...on the words of given lengths and the parity-alternative property in some sets of such words are established.
This paper is contributed to the combinatorial properties of the MSS sequences, which are the periodic kneading words of quadratic maps defined on a interval. An explicit expression of adjacency ...relations on MSS sequences of given lengths is established.
Inverse limit spaces of one-dimensional continua frequently appear as attractors in dissipative dynamical systems. As such, there has been considerable interest in the topology of these inverse limit ...spaces. In this work we describe the topology of Markov interval maps, and use our results to show that for unimodal interval maps with finite kneading sequences, the kneading sequence and dynamics of the left endpoint determine the topology of the associated inverse limit space.
In the presence of stimulatory concentrations of glucose, the membrane potential of pancreatic beta-cells may experience a transition from periods of rapid spike-like oscillations alternating with a ...pseudo-steady state to spike-only oscillations. Insulin secretion from beta-cells closely correlates the periods of spike-like oscillations. The purpose of this paper is to study the mathematical structure which underlines this transitional stage in a pancreatic beta-cell model. It is demonstrated that the transition can be chaotic but becomes more and more regular with increase in glucose. In particular, the system undergoes a reversed period-doubling cascade leading to the spike-only oscillations as the glucose concentration crosses a threshold. The transition interval in glucose concentration is estimated to be extremely small in terms of the rate of change for the calcium dynamics in the beta-cells. The methods are based on the theory of unimodal maps and the geometric and asymptotic theories of singular perturbations.
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