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Polynomial entropy on the $n$-fold symmetric product and its suspension

Maša Đorić

Abstract

We prove that the polynomial entropy of the induced map $F_n(f)$ on the $n$-fold symmetric product of a compact space $X$ and its suspension are both equal to $nh_{pol}(f)$, when $f:X\to X$ is a homeomorphism with a finite chain recurrent set $\mathcal{CR}(f)$. We also give a lower bound for the polynomial entropy on the suspension, for a homeomorphism $f$ with at least one wandering point, under certain assumptions.

Polynomial entropy on the $n$-fold symmetric product and its suspension

Abstract

We prove that the polynomial entropy of the induced map on the -fold symmetric product of a compact space and its suspension are both equal to , when is a homeomorphism with a finite chain recurrent set . We also give a lower bound for the polynomial entropy on the suspension, for a homeomorphism with at least one wandering point, under certain assumptions.

Paper Structure

This paper contains 7 sections, 15 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomial entropy and coding
  4. Hyperspaces and induced maps
  5. Suspensions
  6. Finiteness of $NW(F_n(f))$
  7. Main results and proofs

Key Result

Theorem 1

Let $X$ be a compact space, $f:X\to X$ a homeomorphism with a finite chain recurrent set and $n\geqslant2$. Then

Theorems & Definitions (26)

  • Theorem
  • Theorem
  • Proposition 1
  • Proposition 2
  • proof
  • Example 3
  • Definition 4
  • Example 5
  • Lemma 6
  • proof
  • ...and 16 more