Abstract
The existence is proved of a topologically transitive (t.t.) homeomorphism U of the space W = Φ × Z of the formU (ϕ, z)=(T,ϕ, z+f (ϕ)) (Φ ε ϕ, z ε Z), where Φ is a complete separable metric space, T is a t.t. homeomorphism of Φ onto itself, Z is a separable banach space, andf is a continuous map: Φ → z. For the special case W = S1×R, Tϕ=ϕ+θ (θ is incommensurable with 2π) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous functionf: S1→R for which the corresponding homeomorphism (1) is t.t. for allθ incommensurable with 2π.
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Literature cited
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Moscow-Leningrad (1949).
D. V. Anosov, “Geodesic flows on closed riemannian manifolds of negative curvature,” Tr. Matem. In-ta AN SSSR,90, 3–210 (1967).
L. G. Shnirel'man, “Example of a transformation of the plane,” Izv. Donskogo Politekhnicheskogo In-ta,14, 64–74 (1930).
A. S. Besicovitch, “A problem on topological transformation of the plane,” Fund. Math.,28, 61–65 (1937).
J. C. Oxtoby, “Note on transitive transformations,” Proc. Nat. Acad. Sci. U. S.,23, No. 8, 443–446 (1937).
J. C. Oxtoby and S. M. Ulam, “Measure preserving homeomorphisms and metrical transitivity,” Ann. Math.,42, 874–920 (1941).
D. V. Anosov and A. B. Katok, “New examples of ergodie diffeomorphisms of smooth manifolds,” Usp. Matem. Nauk,25, No. 4, 173–174 (1970).
D. V. Anosov and A. B. Katok, “New examples in smooth ergodic theory, ergodic diffeomorphisms,” Trudy Mosk. Matem O-va,23, 3–36 (1970).
W. Gottschalk and G. Hedlund, Topological Dynamics, A.M.S. Colloq., Vol.36, New York (1955).
E. A. Sidorov, “Smooth topologically transitive dynamical systems,” Matem. Zametki,4, No. 6, 751–759 (1968).
J. Cassels, Introduction to the Theory of Diophantine Approximation [Russian translation], Moscow (1961).
E. A. Sidorov, “Topologically indecomposable transformations of n-dimensional space,” Volzhsk. Matem. Sb., No. 5, 326–330 (1966).
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Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 441–452, September, 1973.
The author thanks D. V. Anosov for advice and interest in the work.
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Sidorov, E.A. Topological transitivity of cylindrical cascades. Mathematical Notes of the Academy of Sciences of the USSR 14, 810–816 (1973). https://doi.org/10.1007/BF01147461
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DOI: https://doi.org/10.1007/BF01147461