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Generically hereditarily equivalent continua and topological characterization of generic maximal chains of generalized Ważewski dendrites

Bryant Rosado Silva, Benjamin Vejnar

TL;DR

The work introduces generically hereditarily equivalent continua (GHEC) and proves that generalized Ważewski dendrites $W_M$ are GHEC and strongly GHEC, yielding comeager families of subcontinua all ambiently equivalent to each other. It develops a comprehensive framework linking GHEC with generic chain properties (GCHEC, GCGHEC) through Desintegration-type theorems and a careful analysis of the hyperspace $ ext{Cont}(X)$ and the maximal chains MOA$(X)$, with special emphasis on Peano continua and dendrite structure. A central achievement is the topological characterization of generic maximal chains of $W_M$, showing the existence of comeager orbits under the homeomorphism group and ambient equivalence of generic chains, along with an extension to $W_ullet$ via inverse-limit constructions. These results illuminate the interplay between continuum geometry, hyperspace dynamics, and generic phenomena in topological dynamics and universal minimal flows.

Abstract

The notion of hereditarily equivalent continua is classical in continuum theory with only two known nondegenerate examples (arc, and pseudoarc). In this paper we introduce generically hereditarily equivalent continua, i.e. continua which are homeomorphic to comeager many subcontinua. We investigate this notion in the realm of Peano continua and we prove that all the generalized Ważewski dendrites are such. Consequently, we study maximal chains consisting of subcontinua of generalized Ważewski dendrites and we prove that there is always a generic orbit under the homeomorphism group action. As a part of the proof we provide a topological characterization of the generic maximal chain.

Generically hereditarily equivalent continua and topological characterization of generic maximal chains of generalized Ważewski dendrites

TL;DR

The work introduces generically hereditarily equivalent continua (GHEC) and proves that generalized Ważewski dendrites are GHEC and strongly GHEC, yielding comeager families of subcontinua all ambiently equivalent to each other. It develops a comprehensive framework linking GHEC with generic chain properties (GCHEC, GCGHEC) through Desintegration-type theorems and a careful analysis of the hyperspace and the maximal chains MOA, with special emphasis on Peano continua and dendrite structure. A central achievement is the topological characterization of generic maximal chains of , showing the existence of comeager orbits under the homeomorphism group and ambient equivalence of generic chains, along with an extension to via inverse-limit constructions. These results illuminate the interplay between continuum geometry, hyperspace dynamics, and generic phenomena in topological dynamics and universal minimal flows.

Abstract

The notion of hereditarily equivalent continua is classical in continuum theory with only two known nondegenerate examples (arc, and pseudoarc). In this paper we introduce generically hereditarily equivalent continua, i.e. continua which are homeomorphic to comeager many subcontinua. We investigate this notion in the realm of Peano continua and we prove that all the generalized Ważewski dendrites are such. Consequently, we study maximal chains consisting of subcontinua of generalized Ważewski dendrites and we prove that there is always a generic orbit under the homeomorphism group action. As a part of the proof we provide a topological characterization of the generic maximal chain.
Paper Structure (10 sections, 36 theorems, 122 equations, 1 figure)

This paper contains 10 sections, 36 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generically hereditarily equivalent continua
  4. Generalized Ważewski dendrites
  5. Peano GHEC
  6. GHEC and further generic properties
  7. Generic maximal chains of $W_M$
  8. Properties of generic chains of $MOA(W_M)$
  9. Generic chains are homeomorphically equivalent
  10. Another class of equivalent chains of $\operatorname{MOA}(W_\omega)$

Key Result

Corollary 1.4

If $M \subseteq \{3,4,\ldots\} \cup \{\omega\}$, then the action $\operatorname{Homeo}(W_M) \curvearrowright \operatorname{MOA}(W_M)$ given by has a comeager orbit.

Figures (1)

  • Figure 1: Scheme of the process after the induction's base case: In 1 we have $K_1$ represented by the thicker curve, while $K_2$ is represented in 2. In 3 and 4 we have, respectively, the trees $T_{1,1}^2$ and $T_{1,1}^1$.

Theorems & Definitions (73)

  • Definition 1.1
  • Corollary 1.4
  • Theorem 2.1: Boundary Bumping Theorem - nadlercontinuum
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • ...and 63 more