The Nielsen numbers of iterations of maps on infra-solvmanifolds of type $$\mathrm {(R)}$$ ( R ) and periodic orbits

Abstract

We study the asymptotic behavior of the sequence of the Nielsen numbers {N (f k )}, the essential periodic orbits of f and the homotopy minimal periods of f by using the Nielsen theory of maps f on infra-solvmanifolds of type (R). We give a linear lower bound for the number of essential periodic orbits of such a map, which sharpens well-known results of Shub and Sullivan for periodic points and of Babenko and Bogatyǐ for periodic orbits. We also verify that a constant multiple of infinitely many prime numbers occur as homotopy minimal periods of such a map.

Homotopy Minimal Periods for Hyperbolic maps on Infra-Nilmanifolds

“…Remark 3.5. The fact that sp( D * ) has to appear in the expression for N (f k ) was proved in a more general setting in [9].…”

“…This set has been studied extensively, for example, in [1] for maps on the torus, in [12] for maps on nilmanifolds and in [9, 17] for maps on infra-nilmanifolds.…”