Realizing rotation vectors for torus homeomorphisms

Franks, John

doi:10.1090/S0002-9947-1989-0958891-1

Realizing rotation vectors for torus homeomorphisms
HTML articles powered by AMS MathViewer

by John Franks

Trans. Amer. Math. Soc. 311 (1989), 107-115

DOI: https://doi.org/10.1090/S0002-9947-1989-0958891-1

PDF | Request permission

Abstract:

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 \[ \frac {{{F^q}({x_0}) - {x_0}}}{q} = v\] where ${x_0} \in {R^2}$ is any lift of $x$ and $q$ is the least period of $x$.

References

Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0283793
Morton Brown, A new proof of Brouwer’s lemma on translation arcs, Houston J. Math. 10 (1984), no. 1, 35–41. MR 736573
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
Albert Fathi, An orbit closing proof of Brouwer’s lemma on translation arcs, Enseign. Math. (2) 33 (1987), no. 3-4, 315–322. MR 925994
John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 99–107. MR 967632, DOI 10.1017/S0143385700009366
John Franks, A variation on the Poincaré-Birkhoff theorem, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 111–117. MR 986260, DOI 10.1090/conm/081/986260
Hugo Hadwiger and Hans Debrunner, Combinatorial geometry in the plane, Holt, Rinehart and Winston, New York, 1964. Translated by Victor Klee. With a new chapter and other additional material supplied by the translator. MR 0164279

Rotation vectors and entropy for homeomorphisms homotopic to the identity

Rotation sets of toral maps

John C. Oxtoby, Diameters of arcs and the gerrymandering problem, Amer. Math. Monthly 84 (1977), no. 3, 155–162. MR 433333, DOI 10.2307/2319484