Abstract
The uniform approach to the concept of geometric integrability for discrete dynamical systems on invariant plane sets is suggested. Geometric and analytic necessary and sufficient conditions for the geometric integrability of maps on invariant plane sets are proved. The solution of the coexistence problem of periodic orbits periods for these maps is given. Obtained results are applied, in particular, to description of the set of periodic orbits (least) periods of geometrically integrable maps with the quotient which is a symmetric Lorenz map.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. E. Adler, ‘‘Discretizations of the Landau-Lifshits equation,’’ Theor. Math. Phys. 124, 897–908 (2000).
V. E. Adler, A. I. Bobenko, and Yu. B. Suris, ‘‘Discrete nonlinear hyperbolic equations. Classification of integrable cases,’’ Funct. Anal. Appl. 43, 3–17 (2009).
V. S. Afraimovich, V. V. Bykov, and L. P. Shilnikov, ‘‘Attractive nonrough limit sets of Lorenz-attractor type,’’ Tr. Mosk. Mat. Ob-va 44, 150–212 (1982).
L. Alséda, J. Llibre, M. Misiurewicz, and C. Tresser, ‘‘Periods and entropy for Lorenz-like maps,’’ Ann. Inst. Fourier (Grenoble) 39, 929–952 (1989).
D. V. Anosov, ‘‘Structurally stable systems,’’ Proc. Steklov Inst. Math. 169, 61–95 (1986).
D. V. Anosov and Ye. V. Zhuzhoma, ‘‘Nonlocal asymptotic behavior of curves and leaves of laminations on universal coverings,’’ Proc. Steklov Inst. Math. 249 (2), 1–219 (2005).
A. Anušić, H. Bruin, and J. Činč, ‘‘Topological properties of Lorenz maps derived from unimodal maps,’’ J. Differ. Equat. Appl. 26, 1174–1191 (2020).
S. S. Belmesova and L. S. Efremova, ‘‘On the concept of integrability for discrete dynamical systems. Investigation of wandering points of some trace map,’’ in Nonlinear Maps and their Applications: Selected Contributions from the NOMA 2013 International Workshop, Ed. by R. López-Ruiz, D. Fournier-Prunaret, Y. Nishio, and C. Grácio, Vol. 112 of Springer Proceedings in Mathematics and Statistics (Springer, Cham, 2015), pp. 127–158.
E. V. Blinova and L. S. Efremova, ‘‘On \(\Omega\)-blow-ups in the simplest \(C^{1}\)-smooth skew products of interval mappings,’’ J. Math. Sci. (N. Y.) 157, 456–465 (2009).
S. V. Bolotin and V. V. Kozlov, ‘‘Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom,’’ Izv. Math. 81, 671–687 (2017).
N.-B. Dang, R. Grigorchuk, and M. Lyubich, ‘‘Self-similar groups and holomorphic dynamics: Renormalization, integrability, and spectrum,’’ arXiv: 2010.00675v2 (2021).
L. S. Efremova, ‘‘Differential properties and attracting sets of a simplest skew product of interval maps,’’ Sb. Math. 201, 873–907 (2010).
L. S. Efremova, ‘‘Space of \(C^{1}\)-smooth skew products of maps of an interval,’’ Theor. Math. Phys. 164, 1208–1214 (2010).
L. S. Efremova, ‘‘A decomposition theorem for the space of \(C^{1}\)-smooth skew products with complicated dynamics of the quotient map,’’ Sb. Math. 204, 1598–1623 (2013).
L. S. Efremova, ‘‘Dynamics of skew products of maps of an interval,’’ Russ. Math. Surv. 72, 101–178 (2017).
L. S. Efremova, ‘‘Small \(C^{1}\)-smooth perturbations of skew products and the partial integrability property,’’ Appl. Math. Nonlin. Sci. 5, 317–328 (2020).
L. S. Efremova, ‘‘The trace map and integrability of the multifunctions,’’ J. Phys.: Conf. Ser. 990, 012003 (2018).
L. S. Efremova, ‘‘Small perturbations of smooth skew products and Sharkovsky’s theorem,’’ J. Differ. Equat. Appl. 26, 1192–1211 (2020).
L. S. Efremova, ‘‘Periodic behavior of maps obtained by small perturbations of smooth skew products,’’ Discontinuity, Nonlinearity, Complexity 9, 519–523 (2020).
L. S. Efremova and E. N. Makhrova, ‘‘One-Dimensional Dynamical Systems,’’ Rus. Math. Surv. 76 (5), 55 (2021).
R. Grigorchuk and S. Samarakoon, ‘‘Integrable and chaotic systems associated with fractal groups,’’ Entropy 23, 237 (2021).
R. I. Grigorchuk and A. Zuk, ‘‘The lamplighter group as a group generated by a 2-state automaton, and its spectrum,’’ Geom. Dedicata 87, 209–244 (2001).
F. Hofbauer, ‘‘Periodic points for piecewise monotonic transformation,’’ Ergod. Th. Dyn. Syst. 5, 237–256 (1985).
P. E. Kloeden, ‘‘On Sharkovsky’s cycle coexistence ordering,’’ Bul. Austr. Math. Soc. 20, 171–177 (1979).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Graylock Press, Rochester, 1957), Vol. 1.
V. V. Kozlov, ‘‘Tensor invariants and integration of differential equations,’’ Russ. Math. Surv. 74, 111–140 (2019).
V. V. Kozlov and D. V. Treschev, ‘‘Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics,’’ Sb. Math. 207, 1435–1449 (2016).
K. Kuratowski, Topology (Academic Press, New York, 1966), Vol. 1.
D. Rand, ‘‘The topological classification of Lorenz attractors,’’ Math. Proc. Cambridge Phil. Soc. 83, 451–460 (1978).
A. N. Sharkovsky, ‘‘Coexistence of cycles of a continuous mapping of the line into itself,’’ Ukr. Mat. Zh. 16, 61–71 (1964).
A. N. Sharkovskii, ‘‘Coexistence of cycles of a continuous map of the line into itself,’’ Bifurc. Chaos 5, 1263–1273 (1995).
Yu. B. Suris, ‘‘Integrable mappings of the standard type,’’ Funct. Anal. Appl. 23, 74–76 (1989).
A. P. Veselov, ‘‘Integrable maps,’’ Russ. Math. Surv. 46 (5), 1–51 (1991).
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by E. A. Turilova)
Rights and permissions
About this article
Cite this article
Efremova, L.S. Geometrically Integrable Maps in the Plane and Their Periodic Orbits. Lobachevskii J Math 42, 2315–2324 (2021). https://doi.org/10.1134/S1995080221100073
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080221100073