One-sided Shift Research Articles

Smooth Extensions of Bernoulli Shifts

For homographic extensions of the one-sided Bernoulli shift we construct $\sigma$-finite invariant and ergodic product measures. We apply the above to the description of invariant product probability measures for smooth extensions of one-sided Bernoulli

Remarks on permutive cellular automata

We prove that every two-dimensional permutive cellular automaton is conjugate to a one-sided shift with compact set of states.

How little is little enough?

Let $f$ be a continuous map $f:X\to X$ of a metric space $X$into itself. Often the information about the map is presented in thefollowing form: for a finite collection of compact sets $A_1, \ldots,A_n$ it is known which sets have the images containing other sets, andwhich sets are disjoint. We study similar but weaker than usualconditions on compact sets $A_1, \ldots, A_n$ assuming that the commonintersection of all sets $A_1,\ldots, A_n$ is empty (or making evenweaker but more technical assumptions). As we show, this implies thatthe map is chaotic in the sense that it has positive topologicalentropy, and moreover, there exists an invariant compact set on which$f$ is semiconjugate to a full one-sided shift.

TheHausdorff dimension of recurrent sets in symbolic spaces

Let (Σ,σ) be the one-sided shift space on m symbols. Foranyx = (xi)i⩾1∊Σ and positive integer n, defineWe prove that for each pair of numbers α,β∊[0,∞]withα⩽β, the following recurrent sethas Hausdorff dimension one.

On finitely Bernoulli one-sided Markov shifts and their cofiltrations

The problem under consideration is: when is a Markov endomorphism (one-sided shift) $T=T_P$ with transition matrix $P$, isomorphic to a Bernoulli endomorphism $\tilde{T}_\rho$ with an appropriate stationary vector $\rho=\{\rho_i\}_{i\in I}$? An obvious necessary condition is that there exists an independent complement $\delta$ of the measurable partition $T^{-1}\varepsilon$ with $\distr \delta = \rho$. In this case the cofiltration (decreasing sequence of measurable partitions) $\xi(T)=\{T^{-n}\varepsilon\}^{\infty}_{n=1}$ generated by $T$ is finitely isomorphic to the standard Bernoulli cofiltration $\xi(\tilde{T}_{\rho})=\{\tilde{T}^{-n}_{\rho}\varepsilon\}^{\infty}_{n=1}$ and $T$ is called finitely $\rho$-Bernoulli.We show that every ergodic Markov endomorphism $T_P$, which is finitely Bernoulli, can be represented as a skew product over $\tilde{T}_{\rho}$ with $d$-point fibres $(d\in \mathbb{N})$. We compute the minimal $d=d(T_P)$ in these skew-product representations by means of the transition matrix, and obtain necessary and sufficient conditions under which $d(T_P)=1$, i.e. $T_P$ is isomorphic to $\tilde{T}_{\rho}$. The cofiltration $\xi(T)$ of any finitely Bernoulli ergodic Markov endomorphism $T=T_P$ is represented as a $d$-point extension of the standard cofiltration $\xi(\tilde{T}_{\rho})$, and we show that the minimal $d=d_{\xi}(T)$ in these extensions is equal to $d(T)$. In particular, $d(T)=1\Longleftrightarrow d_{\xi}(T)=1$, that is, a Markov endomorphism $T$ is isomorphic to $\tilde{T}_{\rho}$ iff $\xi(T)$ is isomorphic to the Bernoulli cofiltration $\xi(\tilde{T}_{\rho})$.

Decay of Correlations for Non-Hölderian Dynamics. A Coupling Approach

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.

Examples and properties of nonexact ergodic shift measures

We construct examples of nonexact n-to-one shifts. We first construct examples of one-sided shift measures with maximal automorphic factors of any prescribed finite cardinality. Then we give an example of a two-to-one shift with a maximal automorphic factor which is an odometer.

Expansivity, Permutivity, and Chaos for Cellular Automata

We study the dynamics of expansive cellular automata. We prove that a cellular automaton is expansive if and only if it is topologically conjugate to an appropriate one-sided subshift. We define a large class of expansive cellular automata in terms of the permutivity of the local rule on which they are based. We prove that each cellular automaton of this class is topologically conjugate to a one-sided full shift. Since one-sided full shifts are widely recognized as the paradigm of chaotic systems, we conclude that the above mentioned cellular automata are chaotic according to any reasonable definition of chaos. We provide a technique to construct expansive cellular automata which are topologically conjugate to one-sided full shifts but do not belong to the above defined class. Finally, we investigate connections between expansivity, topological transitivity, and surjectivity.

Temperatures Differ Between Fronts and Backs of Shaded and Sunlit Berries

Close-focus pyrometers were employed to measure temperatures among blackberry fruits at sunlit and shaded locations on static and one-sided shift trellises with westward leaning (20 to 25° from vertical) crop support arms. About 85% of the static trellis' fruits were located on its eastern face while essentially all of the shift trellis' berries were westward-oriented. During cloudless weather, east-facing berries at 9:30 am were 5 to 8 °C warmer than either ambient air or berries on western trellis faces. West-facing berries were at or near air temperature until 2 or 3 pm; east-facing berries remained above air temperature until 4:30 or 5 pm. Temperatures at the sunlit sides of healthy, east-facing `Arapaho' berries averaged 8 °C warmer than at shaded sides of the same berries.

Shift endomorphisms and compact Lie extensions

We consider skew-products with an arbitrary compact Lie group, when the base map is a one-sided shift of finite type endowed with an equilibrium state of a Holder continuous function. First we show that the weak-mixing property of the skew-product implies exactness and exponential mixing. Then we address the problem of classification under measure-theoretic isomorphisms. We show that for a generic set of equilibrium states the isomorphism class of the skew-products corresponds essentially to the cohomology classes of the defining skewing function and the isomorphism is essentially a homeomorphism.

Quantum geometry of noncommutative Bernoulli shifts

We construct an example of a noncommutative dynamical system defined over a two dimensional noncommutative differential manifold with two positive Lyapunov exponents equal to ln d each. This dynamical system is isomorphic to the quantum Bernoulli shift on the half-chain with the quantum dynamical entropy equal to 2 ln d. This result can be interpreted as a noncommutative analog of the isomorphism between the classical one-sided Bernoulli shift and the expanding map of the circle and moreover as an example of the noncommutative Pesin theorem.

Rigidity of Continuous Coboundaries

We consider the functional equation FoT−F=f, where T is a measure-preserving transformation and f is a continuous function. We show that if there is an L∞ function F which satisfies this equation, then F is constrained to satisfy a number of regularity conditions, and, in particular, if T is a one-sided Bernoulli shift, then we show that there is a continuous function F satisfying this equation. We show that this is not the case for the two-sided shift. 1991 Mathematics Subject Classification 28D05, 58F11.

Mixing properties of one-dimensional cellular automata

We study a class of endomorphisms on the space of bi-infinite sequences over a finite set, and show that such a map is onto if and only if it is measure-preserving. A class of dynamical systems arising from these endomorphisms are strongly mixing, and some of them even m m -mixing. Some of these are isomorphic to the one-sided shift on Z n \mathbb {Z}_n in both the topological and measure-theoretical sense. Such dynamical systems can be associated to O n \mathcal {O}_n , the Cuntz-algebra of order n n , in a natural way.

Generalized one-sided shifts and maps of the interval

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.

One-Sided Shift Trellising Addresses Harvest, Pest Management and Cultural Problems in Rubus

Static-V trellis increases raspberry yield, but fruiting shoots grow toward its center making harvest difficult. Shading causes early leaf abscission and it favors fungus diseases inside the V. Static training of floricanes and primocanes to opposite sides of a V trellis prevents neither harvest difficulty, nor primocane injury during harvest. In 1988-89 harvest difficulties were reduced by bloom-time shifting of floricanes on a V-trellis. Over 90% of fruiting shoots were oriented to the trellis ' exterior, but primocane shading and early leaf abscission continued. The 'bent fence' trellis was designed and tested in 1989. It shifts floricanes from horizontal orientation to an upright position on one side of a V-shaped trellis, thus retaining outward display of fruiting shoots and achieving unobstructed display of primocanes on the opposite side. Harvest efficiency, disease reduction, accurate deposition of pesticides, avoidance of solar injury (sun scald), and adaptation of mechanical pest control procedures are potentiated by this system. It also reduces impediments to studies of carbohydrate partitioning, photosynthetic efficiency, yield efficiency, and intraplant competition.

Spectral multiplicity of certain contractions

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.

A horseshoe for the doubling operator: Topological dynamics for metric universality

The doubling operator, properly defined on the space of smooth maps on the interval at the boundary of chaos, yields a dynamical system in this function space. Even if one restricts oneself to the space of real analytic maps, there is evidence that the dynamics of the doubling operator contains a horseshoe whose symbolic dynamics is described by the one-sided shift on two symbols. We indicate also how some of the global aspects of this dynamics could be recognized in a physical experiment on the transition to chaos.

Generating special Markov partitions for hyperbolic toral automorphisms using fractals

AbstractWe show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.

On frequency waves

A new type of wave existing in the frequency domain is deduced and named a frequency wave. For a homogeneous medium, we have obtained the approximate and rigorous solutions of a frequency shift equation for plane radiation under some special, one-sided or two-sided frequency shift conditions.

A class of nonparametric sequential tests

A class of nonparametric sequential tests is considered for testing a symmetric density under the hypothesis against a one-sided shift alternative. The test statistic at each observation is the sum of intermediate statistics obtained from the ranks within the most recent m observations, where tn is a fixed ranking size. Excessive ranking of the data can be avoided with a proper choice of tn so that real-time implementation of the sequential rank test is feasible. Approximate expressions for the power and average sample number functions are given. Comparison with existing nonparametric tests is studied. Results show that the ranking size m need not be very large in order that the performance of the proposed test be almost as good as sequential rank tests which require ranking of all the data.