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Properties of the Tent map for decimal fractions with fixed precision

The one-dimensional discrete Tent map is a well-known example of a map whose fixed points are all unstable on the segment [0,1]. This map leads to the positivity of the Lyapunov exponent for the corresponding recurrent sequence. Therefore in a situation of general position, this sequence must demonstrate the properties of deterministic chaos. However if the first term of the recurrence sequence is taken as a decimal fraction with a fixed number “k” of digits after the decimal point and all calculations are carried out accurately, then the situation turns out to be completely different. In this case, first, the Tent map does not lead to an increase in significant digits in the terms of the sequence, and secondly, demonstrates the existence of a finite number of eventually periodic orbits, which are attractors for all other decimal numbers with the number of significant digits not exceeding “k”.


1.Introduction
Interest in deterministic chaos among mathematicians and physicists was accompanied by a huge number of publications after the 70s of the last century [1][2][3][4][5]. A large survey of only one of the important areas of these studies [6] contained 216 references.
Significant interest of cryptologists towards deterministic chaos began, apparently from work [7], where the author gave a symmetric encryption scheme based on ergodic property of chaos.
Deterministic chaos is interesting for cryptologists because on the one hand it looks like a random process, but on the other hand it allows to restore the process accurately if the generating model and the initial conditions are known. A detailed discussion of the prospects and problems of new cryptosystems is devoted to the work [8][9][10][11]. Statistical analysis of the Tent map properties by constructing a four-level discrete model is devoted to work [12].
It is known that when chaotic systems are realized with a finite precision in digital computers, their dynamical properties are often found to be entirely different from the original versions in the continuous setting. To emphasize the essential difference between continuous chaos and digital chaos, the latter is also called pseudo chaos [13]. Similarly, digital chaotic orbits are also called pseudo (chaotic) orbits [14].

2.Mathematical definitions and assertions
The paper [15] reports some findings on a new series of dynamical indicators, which can quantitatively reflect the degradation effects on digital 1D piecewise linear chaotic maps realized with a fixed-point finite precision. To describe the processes taking place with limited accuracy of the representation of numbers, the authors used n-bit fixed-point binary decimals in the form In this paper we consider orbits of a recurrent sequence 2 1234567890 ''"" is a simplest Tent map without any parameter which maps [0,1] into [0,1]. In contrast to [15], it is assumed that the initial point of the orbit can be represented as a decimal fraction containing a given number of digits after the decimal point, and that all calculations related to the mapping (2) are carried out exactly without rounding. This possibility is implemented in mathematical packages in the form of symbolic calculations (for example, in the package "Mathematica").
We introduce several definitions that make it possible to formulate the results most simply. Definition1. The set piecewise linear mapping. It is easy to prove that the number of elements of this set is   If you arrange the elements of the cycle ) ( k k m S in ascending order, then the differences between two adjacent elements will be equal to either . In this case, the following rigid order of alternation is realized: at the beginning of the ordered series, a smaller difference is repeated three times, and then a large difference appears once. Then the situation repeats.

3.Conclusions
The Tent map (1) has the following two important properties. First, this transformation does not increase the number of significant digits for each subsequent transformation. Secondly, this transformation has attractors, which are determined by the number of digits after the decimal point. These properties are proposed to be used in the Baptista's encryption scheme instead of logistic mapping. The advantage of the proposed mapping is that there is no need to partition the interval (0,1) into discrete subsets to fix a number. The drawbacks of the proposed scheme include the need for symbolic calculations.