Spaces admitting topologically transitive maps
by
Anima Nagar
University of Hyderabad
Coauthors: V. Kannan (University of Hyderabad)

In this paper, after observing that on certain topological spaces there are no topologically transitive maps at all, we characterize all those locally compact subspaces of the real line that admit a topologically transitive map.

Examples of transitive maps on the unit circle S¹, the Cantor set K and the compact interval [0, 1] are well known. See [5] or [7]. Some explicit examples of transitive maps on the real line R were constructed in [1], after observing that they must have infinitely many critical points [2].

It is also known already that the following topological spaces admit topologically transitive maps (satisfying some additional properties) see [8]:
Any connected region in Rⁿ. (see Sidorov[11])
The Torus S¹ ×S¹. (see Devaney[5])
The cylinder S¹ ×R. (see Besicovitch[3])
The Banach Space l₁. (see C.J.Read[10])
The square I ×I. (see Xiao-Quan Xu[12])

Therefore the following is one of the natural questions: What are all the subspaces of R that admit topologically transitive maps? We provide a complete answer, among locally compact spaces.

We prove,

Theorem
Let X be any locally compact subspace of R. Then X admits a topologically transitive map if and only if X is homeomorphic to one of the following:

1. Finite Discrete space
2. Finite union of nontrivial compact intervals
3. Finite union of nontrivial noncompact intervals
4. Cantor set C
5. C ×N.

References

[1] Anima Nagar and S.P.Sesha Sai, Some Classes of Transitive maps on R, Jour. of Anal.8(2000), 103 - 111.

[2] Anima Nagar, V. Kannan and S.P.Sesha Sai, Some Properties of Topologically Transitive maps on R, Real Anal. Exchange( to appear).

[3] Besicovitch A. S., A problem on topological transformation of the plane, Fund. Math. 28(1937), 61 - 65.

[4] Block L. S. and Coppel W. A., Dynamics in one dimension, Lecture Notes in Mathematics, 1513, Springer Verlag, 1992.

[5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989.

[6] Hocking J. S. and Young G. S., Topology, Addison-Wesley Publishing Company, Reading, Mass., 1961.

[7] R. A. Holmgren, A first course in Discrete Dynamical Systems, Springer-Verlag, 1996.

[8] Kolyada S and Snoha L, Some aspects of topological transitivity - a survey, Grazer Math. Ber., 334(1997), 3 - 35.

[9] S. Mazurkiewicz and W. Sierpinski, Contribution a la topologic des ensembles demombrables, Fund. Math. 1(1920), 17 -27.

[10] C. J. Read, A solution to the invariant subspace problem on the space l₁ , Bul. London Math. Soc., 17(1985), 305 - 317.

[11] Ye. A. Sidorov, Smooth topologically transitive dynamical systems, Mat. Zametki 4 (1968), no. 6, 441 - 452.

[12] Xiao-Quan Xu, Explicit transitive automorphisms of the closed unit square, Proc. Amer. Math. Soc. 109(1990), 571 - 572.

Date received: January 25, 2001