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Discrete and Continuous Dynamical Systems - S

April 2017 , Volume 10 , Issue 2

Issue on diffusion on fractals and non-linear dynamics

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Preface: Diffusion on fractals and non-linear dynamics
Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher and Tony Samuel
2017, 10(2): i-iv doi: 10.3934/dcdss.201702i +[Abstract](2937) +[HTML](1191) +[PDF](140.8KB)
Spectral notions of aperiodic order
Michael Baake and Daniel Lenz
2017, 10(2): 161-190 doi: 10.3934/dcdss.2017009 +[Abstract](3410) +[HTML](173) +[PDF](772.0KB)

Various spectral notions have been employed to grasp the structure and the long-range order of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and present generalisations in the form of further examples and remarks.

Stability of nonlinear waves: Pointwise estimates
Margaret Beck
2017, 10(2): 191-211 doi: 10.3934/dcdss.2017010 +[Abstract](3439) +[HTML](132) +[PDF](769.9KB)

This is an expository article containing a brief overview of key issues related to the stability of nonlinear waves, an introduction to a particular technique in stability analysis known as pointwise estimates, and two applications of this technique: time-periodic shocks in viscous conservation laws [3] and source defects in reaction diffusion equations [1, 2].

A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy
Kristin Dettmers, Robert Giza, Rafael Morales, John A. Rock and Christina Knox
2017, 10(2): 213-240 doi: 10.3934/dcdss.2017011 +[Abstract](2849) +[HTML](160) +[PDF](1163.9KB)

The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of \begin{document}$\mathbb{R}$\end{document} which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.

Random interval diffeomorphisms
Masoumeh Gharaei and Ale Jan Homburg
2017, 10(2): 241-272 doi: 10.3934/dcdss.2017012 +[Abstract](3789) +[HTML](134) +[PDF](1117.2KB)

We consider a class of step skew product systems of interval diffeomorphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present in a simplified setting intriguing phenomena of intermingled basins, master-slave synchronization and on-off intermittency. We provide a self-contained discussion of these phenomena.

Homoclinic tangencies to resonant saddles and discrete Lorenz attractors
Sergey Gonchenko and Ivan Ovsyannikov
2017, 10(2): 273-288 doi: 10.3934/dcdss.2017013 +[Abstract](4912) +[HTML](108) +[PDF](733.1KB)

We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map \begin{document}$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$\end{document} which, as known [14], exhibits discrete Lorenz attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of systems possessing discrete Lorenz attractors near the original diffeomorphism.

A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket
Kumiko Hattori, Noriaki Ogo and Takafumi Otsuka
2017, 10(2): 289-311 doi: 10.3934/dcdss.2017014 +[Abstract](3140) +[HTML](129) +[PDF](305.6KB)

We show that the ‘erasing-larger-loops-first’ (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpiński gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the ‘standard’ self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent \begin{document}$ν$\end{document} governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension \begin{document}$1/ν $\end{document}, which is strictly greater than \begin{document}$1$\end{document}.

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors
Gerhard Keller
2017, 10(2): 313-334 doi: 10.3934/dcdss.2017015 +[Abstract](4266) +[HTML](139) +[PDF](761.7KB)

Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins [1, 5, 16, 30]. To quantify the degree of intermingledness the uncertainty exponent [23] and the stability index [29, 20] were suggested and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.

A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory
Marc Kesseböhmer and Sabrina Kombrink
2017, 10(2): 335-352 doi: 10.3934/dcdss.2017016 +[Abstract](3489) +[HTML](150) +[PDF](648.2KB)

We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.

Derivatives of slippery Devil's staircases
Jun-Jie Miao and Sara Munday
2017, 10(2): 353-365 doi: 10.3934/dcdss.2017017 +[Abstract](3194) +[HTML](157) +[PDF](240.5KB)

In this paper we first give a survey of known results on the derivative of slippery Devil's staircase functions, that is, functions that are singular with respect to the Lebesgue measure and strictly increasing. The best known example of such a function is the Minkowski question-mark function, which was proved to be singular by Salem, in a paper which introduced some other constructions of singular functions. We describe all of these examples. Also we consider various generalisations of the Minkowski question-mark function, such as $α$-Farey-Minkowski functions. These examples all arise from one-dimensional dynamics. A few open questions and suggestions for filling minor gaps in the literature are proposed. Finally, we go back to ordinary Devil's staircases (i.e. non-decreasing singular functions) and discuss work done in that setting with the more general Hölder derivatives, and consider the outlook to extend those results to the strictly increasing situation.

Variational principles for the topological pressure of measurable potentials
Marc Rauch
2017, 10(2): 367-394 doi: 10.3934/dcdss.2017018 +[Abstract](3026) +[HTML](132) +[PDF](491.9KB)

We introduce notions of topological pressure for measurable potentials and prove corresponding variational principles. The formalism is then used to establish a Bowen formula for the Hausdorff dimension of cookie-cutters with discontinuous geometric potentials.

2020 Impact Factor: 2.425
5 Year Impact Factor: 1.490
2020 CiteScore: 3.1

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