Markov Characteristics for IFSP and IIFSP
Subject Areas : Transactions on Fuzzy Sets and Systems
Nan Jiang
1
(
School of Science, Xi’an Shiyou University, Xi’an, China
)
Wei Li
2
(
Institute for Advanced Studies in The History of Science, Northwest University, Xi’an, China.
)
Fei Li
3
(
School of Science, Xi’an Shiyou University, Xi’an, China.
)
Juntao Wang
4
(
School of Science, Xi’an Shiyou University, Xi’an, China.
)
Keywords: Fractal, Markov process, Iterated function system, Probability,
Abstract :
As the research object of modern nonlinear science, a fractal theory has been an important research content for scholars since it comes into the world. Moreover, iterated function system (IFS) is a significant research object of fractal theory. On the other hand, the Markov process plays an important role in the stochastic process. In this paper, the iterated function system with probability(IFSP) and the infinite function system with probability(IIFSP) are investigated by using interlink, period, recurrence and some related concepts. Then, some important properties are obtained, such as: 1. The sequence of stochastic variable $\{\zeta_{n},(n\geq 0)\}$ is a homogenous Markov chain. 2. The sequence of stochastic variable $\{\zeta_{n},(n\geq 0)\}$ is an irreducible ergodic chain. 3. The distribution of transition probability $ p_{ij}^{(n)}$ based on $n\rightarrow\infty $ is a stationary probability distribution. 4. The state space can be decomposed of the union of the finite(or countable) mutually disjoint subsets, which are composed of non-recurrence states and recurrence states respectively.
[1] L. Andrzej, A variational principle for fractal dimensions, Nonlinear Analysis, 64 (2006) 618-628.
[2] M. F. Barnsley, Fractal everywhere, Academic press, New York, (1988).
[3] C. Escribano, M. A. Sastre, E. Torrano, A fixed point theorem for moment matrices of self-similar measures, Journal of Computational and Applied Mathematics, 207 (2007) 352-359.
[4] J. M. Fraser, Fractal geometry of Bedford-McPsillen carpets, Thermodynamic Formalism, (2021) 495-516.
[5] E. Guariglia, Harmonic sierpinski gasket and applications, Entropy, 20 (2018) 714.
[6] J. E. Hutchinson, Fractal and self-similarity, Indiana Univ Math., 30 (1981) 713-747.
[7] M. S. Islam, M. J. Islam, Dynamics of fractal in Euclidean and measure spaces: a recent study, Newest Updates in Physical Science Research, 13 (2021) 43-60.
[8] N. Jiang, Brief history of fractal, Publishing House of Electronics Industry, Bei Jing, (2020).
[9] N. Jiang, N. N. Ma, Probability measure and dirac measure based on hyperbolic IFSP, Journal of Jilin University (Science Edition), 55 (2017) 581-586.
[10] N. Jiang, A. J. Qu, The generation of the concept of fractional dimension, Studies in Philosophy of Science and Technology, 37 (2020) 87-93.
[11] J. Joanna, How to construct asymptotically stable iterated function system, Statistics and Probability Letters, (2008) 1-7.
[12] W. L. John, E. W. Tad, Linear Markov iterated function system, Computers and Graphics, 14 (1990) 343-353.
[13] A. M. Jurgens, P. C. James, Shannon entropy rate of hidden Markov processes, Journal of Statistical Physics, 183 (2021) 1-18.
[14] F. Kenneth, Fractal Geometry Mathematical Foundations And Applications, Thomson Press (India) Ltd, London, (1990).
[15] K. Leniak, S. Nina, Iterated function system enriched with symmetry, Constructive Approzetamation, (2021) 1-21.
[16] I. D. Morris, S. Cagri, A strongly irreducible affine iterated function system with two invariant measures of mazetamal dimension, Ergodic Theory and Dynamical System, 41 (2021) 3417-3438.
[17] Z. Sha, H. J. Ruan, Fractal and fitting, Zhejiang University Press, Hang zhou, (2005).
[18] D. W. Stroock, An introduction to Markov processes, Springer Science & Business Media, Berlin, (2013).
[19] T. Szarek, Z. Anna, The central limit theorem for iterated function system on the circle, Mosc. Math. J., 21 (2021) 175-190.
[20] Z. Tian, C. Y. Qing, et al, Stochastic process and its application, Science Press, Bei Jing, (2007).
[21] D. L. Torre, et al, Iterated function system with place-dependent probabilities and the inverse problem of measure approzetamation using moments, Fractals, 26 (2018) 1850076.
[22] K. Weihrauch, Computability on the probability measures on the Bore1 sets of the unit interval, Theoretical Computer Science, 219 (1999) 421-437.
[23] E. H. Ye, D. P. Zhang, Probability theory and stochastic process, Science Press, Bei Jing, (2005).
