Entropy conjecture for continuous maps of nilmanifolds

Topology and its Applications

Lee

2015

We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Finally we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type (R) whenever all the Reidemeister numbers of iterates of the map are finite. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type (R).

Section: Topological Entropy and The Radius Of Convergencementioning

confidence: 97%

The Nielsen and Reidemeister numbers of maps on infra-solvmanifolds of type (R)

Topology and its Applications

Lee

2015

“…There is a conjectural inequality h(f ) ≥ log(sp(f )) raised by Shub [36]. This conjecture was proven for all maps on infra-solvmanifolds of type (R), see [31,32] and [12]. Now we can state about relations between N ∞ (f ), λ(f ) and h(f ).…”

Section: Radius Of Convergence Of N F (Z)mentioning

confidence: 97%

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

J. Fixed Point Theory Appl.

Lee

2018

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.

“…There is a conjectural inequality h(f ) ≥ log(sp(f )) raised by Shub [44]. This conjecture was proven for all maps on infra-solvmanifolds of type (R), see [39,40] and [15]. Consider a continuous map f on a compact connected manifold M , and consider a homomorphism ϕ induced by f of the group Π of covering transformations on the universal cover of M .…”

Section: Radius Of Convergence Of R ϕ (Z)mentioning

confidence: 99%

The Nielsen and Reidemeister theories of iterations on infra-solvmanifolds of type (𝑅) and poly-Bieberbach groups

Lee²

2016

Dynamics and Numbers

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 develop the Reidemeister theory for the iterations of any endomorphism ϕ on an arbitrary group and study the asymptotic behavior of the sequence of the Reidemeister numbers {R(ϕ k )}, the essential periodic [ϕ]-orbits and the heights of ϕ on poly-Bieberbach groups.