Finite Number Of Symbols Research Articles

Some factors of nonsingular Bernoulli shifts

We give elementary constructions of factors of nonsingular Bernoulli shifts. In particular, we show that all nonsingular Bernoulli shifts on a finite number of symbols which satisfy the Doeblin condition have a factor that is equivalent to an independent and identically distributed system. We also prove that there are type-III:1 Bernoulli shifts of every possible ergodic index, answering a question of Danilenko and Lemanczyk (Ergodic Theory Dynam. Systems, 39(12):3292-3321, 2019).

Expansive dynamics on locally compact groups

AbstractLet $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

Single-carrier System with Time-based Receive Transformation under Intersymbol Interference

In this paper, we study the single-carrier system with time-based receive transformation (TRT) under intersymbol interference (ISI). We consider the single-carrier single-input-single-output (SISO) system where the receiver uses the signals received during finite number of symbol periods to detect one data frame consisting of finite number of symbols. We show that the single-carrier system with TRT can completely eliminate the effect of the inter-frame ISI (IFI) between different data frames so that the diversity provided by the intra-frame ISI within the same data frame can be fully utilized to maximize the achievable rate. We also show that the achievable rate of the single-carrier system with TRT can be larger than the ISI-free SISO channel capacity and this advantage increases with the number of symbols per frame and/or the number of symbols that ISI affects.

Abstract similarity, fractals and chaos

<p style='text-indent:20px;'>A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.

Lyapunov optimization for non-generic one-dimensional expanding Markov maps

For a non-generic, yet dense subset of$C^{1}$expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new$C^{1}$perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

Full shifts and irregular sets

By Birkhoff´s ergodic theorem, the set of points X for which the Birkhoff averages of a continuous function diverge has zero measure with respect to any finite invariant measure. Thus, at least from the point of view of ergodic theory, this set could not be smaller. Nevertheless, it can be large from other points of view. For example, for subshifts with the weak specification property, we showed recently, that X is residual whenever it is nonempty (it is a simple exercise to show that X is dense whenever it is nonempty). The main purpose of this note is to convey in the simplest possible manner the proof of our result in the particular case of the full shift on a finite number of symbols. This has the advantage of avoiding some accessory technicalities that are necessary in the general case. In fact, we consider also the more general case when the set of accumulation points of the Birkhoff averages of a continuous function is prescribed closed interval and we show that it is residual whenever it is nonempty.

Recurrence in pairs

AbstractWe introduce and study the notion of weak product recurrence. Two sufficient conditions for this type of recurrence are established. We deduce that any point with a dense orbit in either the full one-sided shift on a finite number of symbols or a mixing subshift of finite type is weakly product recurrent. This observation implies that distality does not follow from weak product recurrence. We have therefore answered, in the negative, a question posed by Auslander and Furstenberg.

A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given.

Chaos in singular impulsive O.D.E.

It is well-known [l] that the existence of a transversal homoclinic orbit of a diffeomorphism implies chaos, i.e. the diffeomorphism possesses a compact domain which is invariant for some of its iterates and such that the dynamics of such an iterate on it is homeomorphic to the Bernoulli shift on a finite number of symbols. However, the location of transversal homoclinic points is generally difficult. One of the most common approaches is a perturbation method based on the derivation of the so-called Melnikov functions [2], i.e. we look for a transversal homoclinic point of a perturbed diffeomorphism when the unperturbed diffeomorphism has a known dynamics. Analytical methods, based on the Lyapunov-Schmidt procedure, are presented in [3-51 for periodically, regularly perturbed autonomous ordinary differential equations (o.d.e.), and in [6-81 for diffeomorphisms. Regularly perturbed impulsive o.d.e. are studied in [8]. Singular perturbation problems are studied analytically in [9, lo] for o.d.e., and in [l 11 for impulsive o.d.e. In all these papers it is assumed that the so-called reduced o.d.e. has a homoclinic orbit. In particular, the following impulsive o.d.e. is studied in [ll]

Computer assisted proof of chaos in the Rössler equations and in the Hénon map

We introduce horseshoe-type mappings which are geometrically similar to Smale's horseshoes. For such mappings we prove by means of the fixed point index the existence of chaotic dynamics - the semi-conjugacy to the shift on a finite number of symbols. Our theorem does not require any assumptions concerning derivatives, it is a purely topological result. The assumptions of our theorem are then rigorously verified by computer assisted computations for the classical Hénon map and for classical Rössler equations.

Scientific Laws that are neither Deterministic nor Probabilistic

The purpose of this paper is to investigate the sorts of laws or hypotheses that systems may be explained by. We picture the scientist as performing a set of experiments (possibly infinite in number) and recording the results, and from these trying to determine the sort of mechanism that is involved. For example, he might be observing a system of particles and recording their positions (to a specified degree of accuracy) every second. Since the data received can be written down, it can be coded as a sequence of o's and I's. The scientific task, as idealised in this paper, is to find a hypothesis, finite in length, which explains this sequence. These hypotheses are to be framed in a particular language with only a finite number of symbols. This paper will take a mathematical rather than a philosophical approach in that there will be a somewhat ruthless disregard of linguistic subtleties, and a good deal of legislation in order to make the ideas precise. The definition of 'explanation' given is essentially framed for probabilistic laws, and thus also for deterministic ones, since these are merely a special case.

RESEARCH NOTE / THE MAP AS INFORMATION CHANNEL: IGNORANCE BEFORE AND AFTER LOOKING AT A CHOROPLETH MAP

In this note, elementary concepts of information theory are applied to develop a measure of the information conveyed to a map reader by a map. For concreteness, a choropleth map is discussed though the framework is applicable to any map with a finite number of symbols colored in a finite number of ways. The measure takes account of prior and posterior ignorance of the map reader, summarized in a set of discrete subjective probability distributions. The framework presented permits empirical estimation of information content and map-reading performance. More important, from the viewpoint of cartographic design, is the potential for incorporating psychophysical relationships between subjective probabilities and map symbolism in order to maximize information conveyed. Cette étude démontre les concepts élémentaires de la théorie de l'information utilisés dans le but d'en arriver à connaître le degré de renseignements transmis au lecteur de la carte. Afin de donner un exemple concret, une carte établie selon la méthode statistique a été utilisée, bien que la méthode d'approche de cette étude peut s'appliquer à toute carte qui présente un nombre limité de symboles colorés d'un nombre limité de manières. Ce degré de renseignements tient compte des connaissances du lecteur avant et après son étude de la carte; le tout réparti suivant des probabilités admises. La méthode d'approche présentée permet une évaluation empirique des renseignements fournis et des résultats obtenus par la lecture de la carte. Plus important encore, considéré au point de vue de la représentation cartographique, est la possibilité d'établir des rapports psychophysiques entre les probabilités admises et le symbolisme cartographique en vue de porter l'information transmise au maximum.

A note on the strong converse of the coding theorem for the general discrete finite-memory channel

The strong converse for the discrete memoryless channel was proved by the author (Wolfowitz, 1957). (The result actually proved is stronger than the strong converse because of the O(~¢/n) term in the exponent). Subsequently the author (1958) and Feinstein (1959) independently gave the capacity C of a discrete finite-memory channel, and proved the strong converse of the coding theorem for the special discrete finitememory channel studied (Wolfowit~, 1957, 1958). In the present note we prove the strong converse for the general discrete finite-memory channel. Thus our result includes t ha t of Wolfowitz (1958) and Feinstein (1959) as a special case. The proof is a slight modification of the proof of Wolfowitz (1958), whose notation and definitions are hereby assumed. For a definition of the capacity C see (Wolfowitz, 1958) or (Feinstein, 1959); for a definition of the general discrete finite-memory channel see (Feinstein, 1959) or (Feinstein, 1958, p. 90). We shall assume without essential loss of generality that both the input and output alphabets consist of two letters, say 0 and l; extension to the case where each alphabet contains any finite number of symbols is trivial. Any sequence of n zeros or ones will be called an n-sequence. A code (N, X) is a set