Total views : 45
The Dynamics of 1-Step Shifts of Finite Type Over Two Symbols
A 1-step shift of finite type over two symbols is a collection of sequences over symbols 0 and 1 with some constrains. The constrains are identified by a set of forbidden blocks which are not allowed to appear in any sequences in the space. The space is of finite type since the number of forbidden blocks is finite and it is of 1-step type since the forbidden blocks are of length of 2. The aim of this paper is to look at the chaotic behaviour of 1-step shift of finite type by considering all spaces of it type. We found that there are six different 1-step shifts of finite type which exhibits totally different dynamics behaviour. We explain the dynamics of each space and then discuss on the difference of the dynamic properties between these spaces. Two of them are chaotic in the sense of Devaney. However the two chaotic shift spaces have totally different behaviour where one of them has trivial dynamics. The other four spaces are not chaotic but, they have some interesting behaviour to be highlighted. It turns out that some of the non-chaotic shift spaces satisfy some chaotic properties.
Blending, Devaney Chaos, Locally Everywhere Onto, Mixing, Shift of Finite Type.
- Assaf IV D, Gadbois S. Definition of chaos. American Mathematical Monthly (letters). 1992; 99(9): 865.
- Banks J, Brooks J, Cairns G, Davis G, Stacey P. On Devaney’s definition of chaos. Mathematical Association of America.1992; 99(4):332-34.
- Cranell A. The role of transitivity in Devaney’s definition of chaos. American Mathematical Monthly. 1995; 102(9):788-93.
- Devaney RL. USA: Perseus Books Publishing: First course in chaotic dynamical systems theory and experiment.1948.
- Devaney RL. Inc., Menlo Park, CA: Benjamin/Cummings Publishing Co.: An introduction to chaotic dynamical systems.1986.
- Devaney RL. USA: Westview Press: An introduction to chaotic dynamical systems. 2003.
- Good C, Knight R, Raines B. Non hyperbolic one-dimensional invariant sets with a countable infinite collection of in homogeneities. Fundamenta Mathematicae. 2006; 192(3):267-89.
- Lardjane S. On some stochastic properties in Devaney’s chaos. Chaos, Solitons & Fractals. 2006; 28:668-72.
- Li TJ, Yorke JA. Period three implies chaos. American Mathematical Monthly.1975; 82(10):985-92.
- Lind D, Marcus B. United Kingdom: The Press Syndicate Of The University Of Cambridge: An introduction to symbolic dynamics and coding. 1995.
- Sabbaghan M, Damerchiloo H. A note on periodic points and transitive maps. Mathematical Sciences Quarterly Journal. 2011; 5(3):259-66.
- Stuart K. Antichaos and adaption. Scientific American.1991; 265(5):78-84.
- There are currently no refbacks.
This work is licensed under a Creative Commons Attribution 3.0 License.