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A survey on the growth rate inequality for sphere endomorphisms

Juliana Xavier

Abstract

We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context.

A survey on the growth rate inequality for sphere endomorphisms

Abstract

We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context.
Paper Structure (17 sections, 24 theorems, 11 equations, 2 figures)

This paper contains 17 sections, 24 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $C^1$ scenario
  3. annulus maps
  4. Semiconjugacies.
  5. Nielsen Theory.
  6. Lefschetz index.
  7. thurston maps with parabolic orbifolds
  8. topological polynomials
  9. indecomposable continua
  10. A sphere branched covering supporting a completely invariant indecomposable continuum.
  11. The Riemann-Hurwitz formula.
  12. examples
  13. Recurrence and periodic orbits
  14. A Kan-like example without periodic points.
  15. An annulus covering of degree $d$ that is not semiconjugate to $z^d$ acting on $S^1$.
  16. ...and 2 more sections

Key Result

Theorem 2.1

If $f:S^2\to S^2$ is a $C^1$, latitude preserving endomorphism of degree 2, then for each $n$, $f^n$ has at least $2^n$ fixed points.

Figures (2)

  • Figure 1: Julia set for $f(z) = z^2 - 0.110 + 0.6557i$
  • Figure 2:

Theorems & Definitions (35)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • ...and 25 more