Topological Conjugacy and Symbolic dynamics of the one dimensional map
DOI:
https://doi.org/10.14738/aivp.95.10762Keywords:
Symbolic dynamics, Symbol space, Shift amp, Topological conjugacy, semi-conjugacyAbstract
In the study of nonlinear physical systems, one encounters possibly random or chaotic behavior, although the techniques may be perfectly deterministic. Topological conjugacy is essential in the study of iterated functions and, more generally, dynamical systems. Our goal is to show that conjugacy (topological) and symbolic dynamics are a distinguished combination of tools of dynamical systems. This paper investigates conjugacy between different one-dimensional maps and discusses semi-conjugacy between doubling maps and shift maps. Finally, we discuss employing techniques from symbolic dynamics to one-dimensional maps, which means how extended dynamics work for one dimensional map.
References
Hadamar J. Les surfaces a courbures opposes et leur ligues geodesiques, J. M. P. A., 5, (1898), 27-73.
Yuri Lima, Symbolic dynamics for one dimensional map with nonuniform expansion, Elsevier, Ann. I. H. Poincare-AN 37, 727-755, (2020).
H. R. Biswas and Monirul Islam, Shift Map and Cantor Set of Logistic Function, IOSR Journal of Mathematics, 16(3),01-08, (2020).
R.L Devany, An introduction to Chaotic Dynamical Systems, 2nd Edition, Addison-Wesley, Redwood City, CA, (1989).
Biswas, H. R., and Md. Shahidul Islam, Chaotic features of the forward Shift Map on the Generalized m-Symbol Space, Journal of Applied Mathematics and Computation, 4 (3): 104-112, (2020).
Clark Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos" ISBN: 0-8493-8493-1.
Brin, M., Stuck, G., Introduction to Dynamical Systems, Cambridge University Press, New York, (2002).
Lind, D., Marcus, B. An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, New York, (1995).
Avrutin V., Schanz M. On the fully developed band count adding scenario. Nonlinearity, 21, 1077-1103 ( 2008).
Avrutin V., Zhusubaliyev Z. T., Mosekilde E. Border collisions inside the stability domain of a fixed point. Physica D, 321(322), 1-15 (2016).
Ju H., Shao H., Choe Y., and Shi Y., Conditions for maps to be topologically conjugate or semi-conjugate to subshifts of finite type and criteria of chaos. Dynamical Systems, International Journal, 31(4), 2016.
Devaney, R.L. An introduction to chaotic dynamical systems, 2nd edition, NewYork: Addison –Wesley, Redwood City, CA (1989).
Osipenko G. Dynamical Systems, Graphs, and Algorithms. Lecture Notes in Mathematics, Heidelberg, Germany, 300 (2007).
S. Ghosh, Chaotic Dynamical Systems, M.Sc. thesis, IIT Hyderabad, (2015).
V. Aleks1. Hadamar J. Les surfaces a courbures opposes et leur ligues geodesiques, J. M. P. A., 5, (1898), 27-73.
Yuri Lima, Symbolic dynamics for one dimensional map with nonuniform expansion, Elsevier, Ann. I. H. Poincare-AN 37, 727-755, (2020).
H. R. Biswas and Monirul Islam, Shift Map and Cantor Set of Logistic Function, IOSR Journal of Mathematics, 16(3),01-08, (2020).
R.L Devany, An introduction to Chaotic Dynamical Systems, 2nd Edition, Addison-Wesley, Redwood City, CA, (1989).
Biswas, H. R., and Md. Shahidul Islam, Chaotic features of the forward Shift Map on the Generalized m-Symbol Space, Journal of Applied Mathematics and Computation, 4 (3): 104-112, (2020).
Clark Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos" ISBN: 0-8493-8493-1.
Brin, M., Stuck, G., Introduction to Dynamical Systems, Cambridge University Press, New York, (2002).
Lind, D., Marcus, B. An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, New York, (1995).
Avrutin V., Schanz M. On the fully developed band count adding scenario. Nonlinearity, 21, 1077-1103 ( 2008).
Avrutin V., Zhusubaliyev Z. T., Mosekilde E. Border collisions inside the stability domain of a fixed point. Physica D, 321(322), 1-15 (2016).
Ju H., Shao H., Choe Y., and Shi Y., Conditions for maps to be topologically conjugate or semi-conjugate to subshifts of finite type and criteria of chaos. Dynamical Systems, International Journal, 31(4), 2016.
Devaney, R.L. An introduction to chaotic dynamical systems, 2nd edition, NewYork: Addison –Wesley, Redwood City, CA (1989).
Osipenko G. Dynamical Systems, Graphs, and Algorithms. Lecture Notes in Mathematics, Heidelberg, Germany, 300 (2007).
S. Ghosh, Chaotic Dynamical Systems, M.Sc. thesis, IIT Hyderabad, (2015).
V. Alekseyev and M, Lakobson, Symbolic Dynamics and Hyperbolic Dynamical Systems, Physics Reports 75, 287-325, (1981).
P. Glendinning, Hyperbolicity of the Invariant set for the Logistic map with μ>4, Nonlinear Analysis, 47, 3323-3332 (2001).
W.Parry, Symbolic Dynamics and Transformation of the unit interval, Tran. Amer. Math. Society, 122(2), 368-378, (1966).
Richard A. Holmgren, A First Course in Discrete Dynamical Systems, 2nd Edition, Springer-Verlag New York, 1996.
Syahida C. D. K., Chaotic Dynamical Systems, e-these repository, University of Birmingham Research Archive 2011.
eyev and M, Lakobson, Symbolic Dynamics and Hyperbolic Dynamical Systems, Physics Reports 75, 287-325, (1981).
P. Glendinning, Hyperbolicity of the Invariant set for the Logistic map with μ>4, Nonlinear Analysis, 47, 3323-3332 (2001).
W.Parry, Symbolic Dynamics and Transformation of the unit interval, Tran. Amer. Math. Society, 122(2), 368-378, (1966).
Richard A. Holmgren, A First Course in Discrete Dynamical Systems, 2nd Edition, Springer-Verlag New York, 1996.
Syahida C. D. K., Chaotic Dynamical Systems, e-these repository, University of Birmingham Research Archive 2011.
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