On the growth rate inequality for periodic points in the two sphere (original) (raw)

On the growth rate inequality for self-maps of the sphere

arXiv (Cornell University), 2023

Let S m = {x 2 0 +x 2 1 +• • •+x 2 m = 1} and P = {x0 = x1 = 0}∩S m. Suppose that f is a self-map of S m such that f −1 (P) = P and |deg(f |P)| < |deg(f)|. Then, the number of fixed points of f n grows at least exponentially with base |d| > 1, where d = deg(f)/deg(f |P) ∈ Z.

A fixed point curve theorem for finite orbits local diffeomorphisms

2021

We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a local biholomorphism F , in dimension 2 with finite orbits, there exists an analytic curve passing through the origin and contained in the fixed point set of some non-trivial iterate of F. As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of local biholomorphisms, with finite orbits, such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they can not be embedded in one parameter groups.

On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

Nonlinearity, 2015

We study bifurcations of area-preserving maps, both orientable (symplectic) and nonorientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.

Harold Bell and the plane fixed point problem

In this paper we present proofs of basic results, including those developed so far by H. Bell, for the plane fixed point problem. Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. We develop a prime end theory through hyperbolic chords in maximal round balls contained in the complement of a non-separating plane continuum X. We define the concept of an outchannel for a fixed point free map which carries the boundary of X minimally into itself and prove that such a map has a unique outchannel, and that outchannel must have variation = −1. We also extend Bell's linchpin theorem for a foliation of a simply connected domain, by closed convex subsets, to arbitrary domains in the sphere.

Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Nonlinearity, 2013

We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.

Action and periodic orbits on annulus

2021

We consider the classical problem of area-preserving maps on annulus A = S × [0, 1] . Let Mf be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism f on A. Given any μ1 and μ2 in Mf , Franks [2][3], generalizing Poincaré’s last geometric theorem (Birkhoff [1]), showed that if their rotation numbers are different, then f has infinitely many periodic orbits. In this paper, we show that if μ1 and μ2 have different actions, even if they have the same rotation number, then f has infinitely many periodic orbits. In particular, if the action difference is larger than one, then f has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.