# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

December 2019 , Volume 12 , Issue 8

Issue on fractal geometry, dynamical systems, and their applications

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2019, 12(8): i-i doi: 10.3934/dcdss.201908i +[Abstract](1803) +[HTML](265) +[PDF](108.2KB)
Abstract:
2019, 12(8): 2211-2220 doi: 10.3934/dcdss.2019142 +[Abstract](2669) +[HTML](738) +[PDF](333.27KB)
Abstract:

We show that shadowing is a generic property for continuous maps on dendrites.

2019, 12(8): 2221-2252 doi: 10.3934/dcdss.2019143 +[Abstract](2613) +[HTML](706) +[PDF](556.98KB)
Abstract:

In this paper, we establish free-probabilistic models \begin{document}$\left( \mathcal{H}(G_{p}),\text{ }\psi _{p}\right)$\end{document} on Hecke algebras \begin{document}$\mathcal{H}(G_{p})$\end{document}, and construct Hilbert-space representations of \begin{document}$\mathcal{H} (G_{p}),$\end{document} preserving free-probabilistic information from \begin{document}$\left( \mathcal{H}(G_{p}),\text{ }\psi _{p}\right) ,$\end{document} for primes \begin{document}$p.$\end{document} From such free-probabilistic structures with representations, we study spectral properties of operators in \begin{document}$C^{*}$\end{document}-algebras generated by \begin{document}$\left\{ \mathcal{H}(G_{p})\right\}_{p:primes}$\end{document}, via their free distributions.

2019, 12(8): 2253-2277 doi: 10.3934/dcdss.2019144 +[Abstract](3013) +[HTML](697) +[PDF](3443.77KB)
Abstract:

We extend the definition of an orbit portrait to the context of non-autonomous iteration, both for the combinatorial version involving collections of angles, and for the dynamic version involving external rays where combinatorial portraits can be realized by the dynamics associated with sequences of polynomials with suitably uniformly bounded degrees and coefficients. We show that, in the case of sequences of polynomials of constant degree, the portraits which arise are eventually periodic which is somewhat similar to the classical theory of polynomial iteration. However, if the degrees of the polynomials in the sequence are allowed to vary, one can obtain portraits with complementary arcs of irrational length which are fundamentally different from the classical ones.

2019, 12(8): 2279-2305 doi: 10.3934/dcdss.2019145 +[Abstract](3300) +[HTML](875) +[PDF](6747.17KB)
Abstract:

A trajectory \begin{document}$\theta_n : = F^n(\theta_0), n = 0,1,2, \dots$\end{document} is quasiperiodic if the trajectory lies on and is dense in some \begin{document}$d$\end{document}-dimensional torus \begin{document}${\mathbb{T}^d}$\end{document}, and there is a choice of coordinates on \begin{document}${\mathbb{T}^d}$\end{document} for which \begin{document}$F$\end{document} has the form \begin{document}$F(\theta) = \theta + \rho\bmod1$\end{document} for all \begin{document}$\theta\in {\mathbb{T}^d}$\end{document} and for some \begin{document}$\rho\in {\mathbb{T}^d}$\end{document}. (For \begin{document}$d>1$\end{document} we always interpret \begin{document}$\bmod1$\end{document} as being applied to each coordinate.) There is an ancient literature on computing the three rotation rates for the Moon. However, for \begin{document}$d>1$\end{document}, the choice of coordinates that yields the form \begin{document}$F(\theta) = \theta + \rho\bmod1$\end{document} is far from unique and the different choices yield a huge choice of coordinatizations \begin{document}$(\rho_1,\cdots,\rho_d)$\end{document} of \begin{document}$\rho$\end{document}, and these coordinations are dense in \begin{document}${\mathbb{T}^d}$\end{document}. Therefore instead one defines the rotation rate \begin{document}$\rho_\phi$\end{document} (also called rotation rate) from the perspective of a map \begin{document}$\phi:T^d\to S^1$\end{document}. This is in effect the approach taken by the Babylonians and we refer to this approach as the "Babylonian Problem": determining the rotation rate \begin{document}$\rho_\phi$\end{document} of the image of a torus trajectory - when the torus trajectory is projected onto a circle, i.e., determining \begin{document}$\rho_\phi$\end{document} from knowledge of \begin{document}$\phi(F^n(\theta))$\end{document}. Of course \begin{document}$\rho_\phi$\end{document} depends on \begin{document}$\phi$\end{document} but does not depend on a choice of coordinates for \begin{document}${\mathbb{T}^d}$\end{document}. However, even in the case \begin{document}$d = 1$\end{document} there has been no general method for computing \begin{document}$\rho_\phi$\end{document} given only the sequence \begin{document}$\phi(\theta_n)$\end{document}, though there is a literature dealing with special cases. Here we present our Embedding continuation method for general \begin{document}$d$\end{document} for computing \begin{document}$\rho_\phi$\end{document} from the image \begin{document}$\phi(\theta_n)$\end{document} of a trajectory, and show examples for \begin{document}$d = 1$\end{document} and \begin{document}$2$\end{document}. The method is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem.

2019, 12(8): 2307-2348 doi: 10.3934/dcdss.2019146 +[Abstract](3236) +[HTML](720) +[PDF](1021.99KB)
Abstract:

We study positive transfer operators \begin{document}$R$\end{document} in the setting of general measure spaces \begin{document}$\left(X,\mathscr{B}\right)$\end{document}. For each \begin{document}$R$\end{document}, we compute associated path-space probability spaces \begin{document}$\left(Ω,\mathbb{P}\right)$\end{document}. When the transfer operator \begin{document}$R$\end{document} is compatible with an endomorphism in \begin{document}$\left(X,\mathscr{B}\right)$\end{document}, we get associated multiresolutions for the Hilbert spaces \begin{document}$L^{2}\left(Ω,\mathbb{P}\right)$\end{document} where the path-space \begin{document}$Ω$\end{document} may then be taken to be a solenoid. Our multiresolutions include both orthogonality relations and self-similarity algorithms for standard wavelets and for generalized wavelet-resolutions. Applications are given to topological dynamics, ergodic theory, and spectral theory, in general; to iterated function systems (IFSs), and to Markov chains in particular.

2019, 12(8): 2349-2364 doi: 10.3934/dcdss.2019147 +[Abstract](2853) +[HTML](637) +[PDF](2129.44KB)
Abstract:

A complex Hadamard matrix \begin{document}$H$\end{document} may be isolated or may lie in a higher-dimensional space of Hadamards. We provide an upper bound for this dimension as the dimension of the center subspace of a gradient flow and apply the Center Manifold Theorem of dynamical systems theory to study local structure in spaces of complex Hadamard matrices. Through examples, we provide several applications of our methodology including the construction of affine families of Hadamard matrices.

2019, 12(8): 2365-2377 doi: 10.3934/dcdss.2019148 +[Abstract](3186) +[HTML](683) +[PDF](400.77KB)
Abstract:

If \begin{document}$S$\end{document} is a semigroup in \begin{document}$\mathbb{R}^n$\end{document} that is not separated by a linear functional, then it is known that the closure of \begin{document}$S$\end{document} is a group. We investigate a similar statement in an infinite dimensional topological vector space \begin{document}$X$\end{document}. We show that if \begin{document}$X$\end{document} is an infinite dimensional Banach space, then there exists a semigroup \begin{document}$S\subset X$\end{document}, not separated by the continuous functionals supported by the closed linear span of \begin{document}$S$\end{document}, for which the closure of the semigroup is not a group. If \begin{document}$X$\end{document} is an infinite dimensional Fréchet space, then the closure of a semigroup that is not separated is always a group if and only if \begin{document}$X$\end{document} is \begin{document}$\mathbb{R}^{\omega}$\end{document}, the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as \begin{document}$\mathbb{R}^{\infty}$\end{document}, the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.

2019, 12(8): 2379-2390 doi: 10.3934/dcdss.2019149 +[Abstract](2911) +[HTML](670) +[PDF](469.23KB)
Abstract:

For every \begin{document}$n ∈ {\mathbb N}$\end{document} we construct orientation preserving planar homeomorphisms \begin{document}$g_n$\end{document} such that \begin{document}$Fix(g_n)=\{0\}$\end{document}, the fixed point index of \begin{document}$g_n$\end{document} at \begin{document}$0$\end{document}, \begin{document}$i_{{\mathbb R}^2}(g_n,0)$\end{document}, is equal to \begin{document}$-n$\end{document} and the stable (respectively unstable) sets of \begin{document}$g_n$\end{document} at \begin{document}$0$\end{document} decompose into exactly \begin{document}$n+1$\end{document} connected branches \begin{document}$\{S_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$\end{document} (resp.\begin{document}$\{U_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$\end{document}) such that:

a) \begin{document}$S_i \cap S_j= \{0\} = U_i \cap U_j$\end{document} for any \begin{document}$i, j ∈ \{1,2, \dotsn+1\}$\end{document} with \begin{document}$i\ne j$\end{document}.

b) \begin{document}$S_i \cap U_j= \{0\}$\end{document} for any \begin{document}$i, j ∈ \{1,2, \dots n+1\}$\end{document}.

c) For every \begin{document}$j ∈ \{1,2, \dots n+1\}$\end{document}, \begin{document}$S_j \setminus\{0\}$\end{document} and \begin{document}$U_j \setminus \{0\}$\end{document} admit translation pseudo-arcs. This means that there exist pseudo-arcs \begin{document}$K_j\subset S_j$\end{document} and points \begin{document}$p_{j\star} , g_n(p_{j\star}) ∈ K_j$\end{document}, such that \begin{document}$g_n(K_j)\cap K_j=\{ g_n(p_{j\star} )\}$\end{document} and

and analogously for \begin{document}$U_j$\end{document}.

We also study the closure of the class of above homeomorphisms in the (complete) metric space of planar orientation preserving homeomorphisms.

2019, 12(8): 2391-2402 doi: 10.3934/dcdss.2019150 +[Abstract](2683) +[HTML](687) +[PDF](408.13KB)
Abstract:

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give a detailed construction of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.

2019, 12(8): 2403-2433 doi: 10.3934/dcdss.2019151 +[Abstract](3365) +[HTML](824) +[PDF](4291.42KB)
Abstract:

Topological mating is a combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map \begin{document}$F$\end{document} on the Riemann sphere. Given a pair of polynomials of the form \begin{document}$z^2+c$\end{document} that are postcritically finite, there is a fast test on the constant parameters to determine whether this map \begin{document}$F$\end{document} exists-but this test does not give a construction of \begin{document}$F$\end{document}. We present an iterative method that utilizes finite subdivision rules and Thurston's algorithm to approximate this rational map, \begin{document}$F$\end{document}. This manuscript expands upon results given by the Medusa algorithm in [9]. We provide a proof of the algorithm's efficacy, details on its implementation, the settings in which it is most successful, and examples generated with the algorithm.

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