×

zbMATH — the first resource for mathematics

Realizing rotation vectors for torus homeomorphisms. (English) Zbl 0664.58028
We consider the rotation set \(\rho\) (F) for a lift F of a homeomorphism f: \(T^ 2\to T^ 2\), which is homotopic to the identity. Our main result is that if a vector v lies in the interior of \(\rho\) (F) and has both coordinates rational, then there is a periodic point \(x\in T^ 2\) with the property that \((F^ q(x_ 0)-x_ 0)/q=v\) where \(x_ 0\in R^ 2\) is any lift of x and q is the least period of x.

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London, 1971. · Zbl 0216.19601
[2] Morton Brown, A new proof of Brouwer’s lemma on translation arcs, Houston J. Math. 10 (1984), no. 1, 35 – 41. · Zbl 0551.57005
[3] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. · Zbl 0397.34056
[4] Albert Fathi, An orbit closing proof of Brouwer’s lemma on translation arcs, Enseign. Math. (2) 33 (1987), no. 3-4, 315 – 322. · Zbl 0649.54022
[5] John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8* (1988), no. Charles Conley Memorial Issue, 99 – 107. · Zbl 0634.58023
[6] John Franks, A variation on the PoincarĂ©-Birkhoff theorem, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 111 – 117.
[7] Hugo Hadwiger and Hans Debrunner, Combinatorial geometry in the plane, Translated by Victor Klee. With a new chapter and other additional material supplied by the translator, Holt, Rinehart and Winston, New York, 1964.
[8] R. MacKay and J. Llibre, Rotation vectors and entropy for homeomorphisms homotopic to the identity, preprint. · Zbl 0699.58049
[9] M. Misiurewicz and K. Ziemian, Rotation sets of toral maps (to appear). · Zbl 0663.58022
[10] John C. Oxtoby, Diameters of arcs and the gerrymandering problem, Amer. Math. Monthly 84 (1977), no. 3, 155 – 162. · Zbl 0355.52007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.