# On the structure of the intersection of two middle third Cantor sets.

• Volume: 39, Issue: 1, page 43-60
• ISSN: 0214-1493

top

## Abstract

top
Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of points in the intersection will be either 2n or 3 · 2n. We also explore the Hausdorff dimension of the intersection of two middle third Cantor sets as the sets are translated across one another. We show that the Hausdorff dimension of the intersection can take on any value from 0 to ln 2 / ln 3; in addition, we show that for each Hausdorff dimension, between 0 and ln 2 / ln 3, there is a dense set of translation parameters for which the intersections have that particular Hausdorff dimension.

## How to cite

top

Davis, Gregory J., and Hu, Tian You. "On the structure of the intersection of two middle third Cantor sets.." Publicacions Matemàtiques 39.1 (1995): 43-60. <http://eudml.org/doc/41222>.

@article{Davis1995,
abstract = {Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of points in the intersection will be either 2n or 3 · 2n. We also explore the Hausdorff dimension of the intersection of two middle third Cantor sets as the sets are translated across one another. We show that the Hausdorff dimension of the intersection can take on any value from 0 to ln 2 / ln 3; in addition, we show that for each Hausdorff dimension, between 0 and ln 2 / ln 3, there is a dense set of translation parameters for which the intersections have that particular Hausdorff dimension.},
author = {Davis, Gregory J., Hu, Tian You},
journal = {Publicacions Matemàtiques},
keywords = {Topología algebraica; Dimensión de Hausdorff; Sistemas dinámicos; Sistemas no lineales; Punto homoclínico; Multiplicidad de intersección; Teoría de bifurcación; homoclinic bifurcations; Cantor set; Hausdorff dimension},
language = {eng},
number = {1},
pages = {43-60},
title = {On the structure of the intersection of two middle third Cantor sets.},
url = {http://eudml.org/doc/41222},
volume = {39},
year = {1995},
}

TY - JOUR
AU - Davis, Gregory J.
AU - Hu, Tian You
TI - On the structure of the intersection of two middle third Cantor sets.
JO - Publicacions Matemàtiques
PY - 1995
VL - 39
IS - 1
SP - 43
EP - 60
AB - Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of points in the intersection will be either 2n or 3 · 2n. We also explore the Hausdorff dimension of the intersection of two middle third Cantor sets as the sets are translated across one another. We show that the Hausdorff dimension of the intersection can take on any value from 0 to ln 2 / ln 3; in addition, we show that for each Hausdorff dimension, between 0 and ln 2 / ln 3, there is a dense set of translation parameters for which the intersections have that particular Hausdorff dimension.
LA - eng
KW - Topología algebraica; Dimensión de Hausdorff; Sistemas dinámicos; Sistemas no lineales; Punto homoclínico; Multiplicidad de intersección; Teoría de bifurcación; homoclinic bifurcations; Cantor set; Hausdorff dimension
UR - http://eudml.org/doc/41222
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.