Conjugacy for certain automorphisms of the one-sided shift via transducers

Distributional chaos occurring on measure center

The new development of a one-sided nonlinear adaptive shift estimation technique (NADSET) is introduced. The background of the algorithm and a brief overview of NADSET are presented. The new technique is applied to the wind parameter estimates from a 2-μm wavelength coherent Doppler lidar system called VALIDAR located in NASA Langley Research Center in Virginia. The new technique enhances wind parameters such as Doppler shift and power estimates in low Signal-To-Noise-Ratio (SNR) regimes using the estimates in high SNR regimes as the algorithm scans the range bins from low to high altitude. The original NADSET utilizes the statistics in both the lower and the higher range bins to refine the wind parameter estimates in between. The results of the two different approaches of NADSET are compared.

The chapter aims for a broad and balanced overview of filmmaking activity in Iceland during the first two decades of the twentieth-first century. It begins by describing an important generational shift that took place early in the century, which was also a mostly one-sided shift in term of gender, as women directors were increasingly marginalized in Icelandic cinema. It also explains how during the aughts Icelandic filmmakers for the first time turned systematically to Hollywood genres and norms, before Hollywood producers themselves increasingly began to bring their offshore productions to Iceland during the teens. Nonetheless, art cinema continued to thrive with many notable international successes, while a documentary revival took place as well.

Endomorphisms of graph algebras

One computes the spectral multiplicity of the direct sum of operators of the following form: a unitary operator U=Ua⊕Us, a contraction T of class C0, a one-sided shift S, and its conjugate S*. Theorem: where δmo=1 if m=0 and δmo=0 it m>0.

The author considers a generalization of the one-sided shift, suitable for describing a certain class of maps in the interval that preserve a Cantor set. The author shows that if such a map is single-valued, it has a finite Markov partition (i.e. that its symbolic dynamics is regular), but if it is multiple-valued, its symbolic dynamics can be an arbitrary context-free language. The scaling properties of sets corresponding to such languages are discussed, and an example is given where the semigroup of scaling operations is infinite-dimensional but finitely describable. The author also discusses the problems of embedding a computationally complex process in one dimension.

Characterization and topological behavior of homomorphism tree-shifts

In the present work, we introduce a partial sequential sampling scheme to develop a sequential rank-based nonparametric test for the identity of two unknown univariate continuous distribution functions against one-sided shift in location occurring at an unknown time point. Our work is motivated by Wolfe (1977) as well as Orban and Wolfe (1980). We provide detailed discussion on asymptotic studies related to the proposed test. We compare the proposed test with a usual rank-based test. Some simulation studies are also presented.

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show these operations are closed under composition. We also study a family of n-ary interleaving operations, one for each n≥1. Given subsets X0,X1,...,Xn−1 of such a shift space, the n-ary interleaving operation produces a set whose elements combine individual elements xi, one from each Xi, by interleaving their symbol sequences in arithmetic progressions (modn). We determine algebraic relations between decimation and interleaving operations and the shift map. We study set-theoretic n-fold closure operations X↦X[n], which interleave decimations of X of modulus level n. A set is n-factorizable if X=X[n]. The n-fold interleaving operations are closed under composition and are idempotent. To each X we assign the set N(X) of all values n≥1 for which X=X[n]. We characterize the possible sets N(X) as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable X, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.

We study a class of dynamical systems in \(L^2\) spaces of infinite products \(X\). Fix a compact Hausdorff space \(B\). Our setting encompasses such cases when the dynamics on \(X = B^\mathbb {N}\) is determined by the one-sided shift in \(X\), and by a given transition-operator \(R\). Our results apply to any positive operator \(R\) in \(C(B)\) such that \(R1 = 1\). From this we obtain induced measures \(\Sigma \) on \(X\), and we study spectral theory in the associated \(L^2(X,\Sigma )\). For the second class of dynamics, we introduce a fixed endomorphism \(r\) in the base space \(B\), and specialize to the induced solenoid \(\mathrm{Sol }(r)\). The solenoid \(\mathrm{Sol }(r)\) is then naturally embedded in \(X = B^\mathbb {N}\), and \(r\) induces an automorphism in \(\mathrm{Sol }(r)\). The induced systems will then live in \(L^2(\mathrm{Sol }(r), \Sigma )\). The applications include wavelet analysis, both in the classical setting of \(\mathbb {R}^n\), and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). But our solenoid analysis includes such hyperbolic systems as the Smale-Williams attractor, with the endomorphism \(r\) there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.

A Layer Rank Test for Ordered Bivariate Alternatives

We consider the classical dynamics given by a one-sided shift on the Bernoulli space of [Formula: see text] symbols. We study, on the space of Hölder functions, the eigendistributions of the Ruelle operator with a given potential. Our main theorem shows that for any isolated eigenvalue, the eigendistributions of such Ruelle operator are dual to eigenvectors of a Ruelle operator with a conjugate potential. We also show that the eigenfunctions and eigendistributions of the Koopman operator satisfy a similar relationship. To show such results we employ an integral kernel technique, where the kernel used is the involution kernel.

We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non Holder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.

We explore the topological full group 〚 G 〛 $\llbracket G\rrbracket $ of an essentially principal étale groupoid G on a Cantor set. When G is minimal, we show that 〚 G 〛 $\llbracket G\rrbracket $ (and its certain normal subgroup) is a complete invariant for the isomorphism class of the étale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup D ( 〚 G 〛 ) $D(\llbracket G\rrbracket )$ is shown to be simple. The étale groupoid G arising from a one-sided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid. For such G, 〚 G 〛 $\llbracket G\rrbracket $ is thought of as a generalization of the Higman–Thompson group. We prove that 〚 G 〛 $\llbracket G\rrbracket $ is of type F ∞, and so in particular it is finitely presented. This gives us a new infinite family of finitely presented infinite simple groups. Also, the abelianization of 〚 G 〛 $\llbracket G\rrbracket $ is calculated and described in terms of the homology groups of G.

Topological behavior, such as chaos, irreducibility, and mixing of a one-sided shift of finite type, is well elucidated. Meanwhile, the investigation of multidimensional shifts, for instance, textile systems, is difficult and only a few results have been obtained so far. This paper studies shifts defined on infinite trees, which are called tree-shifts. Infinite trees have a natural structure of one-sided symbolic dynamical systems equipped with multiple shift maps and constitute an intermediate class between one-sided shifts and multidimensional shifts. We have shown not only an irreducible tree-shift of finite type but also a mixing tree-shift that is chaotic in the sense of Devaney. Furthermore, the graph and labeled graph representations of tree-shifts are revealed so that the verification of irreducibility and mixing of a tree-shift is equivalent to determining the irreducibility and mixing of matrices, respectively. This extends the classical results of one-sided symbolic dynamics. A necessary and sufficient condition for the irreducibility and mixing of tree-shifts of finite type is demonstrated. Most important of all, the examination can be done in finite steps with an upper bound.