## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 231 - 241

### Remarks on non-diagonality conditions for Sierpinski carpets

#### Abstract

We prove the equivalence of three different formulations of non-diagonality for Sierpinski carpets given by Barlow, Bass, Kumagai and Teplyaev [5], by Hino [10] and by Kigami [12]. We also derive some geometric property of Sierpinski carpets from the non-diagonality. As an application, we give a simpler treatment of a criterion stated in Kigami [12, Section 3.4] for the volume doubling property of self-similar measures on Sierpinski carpets.

#### Article information

**Source***Probabilistic Approach to Geometry*, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 231-241

**Dates**

Received: 15 January 2009

Revised: 11 June 2009

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543086321

**Digital Object Identifier**

doi:10.2969/aspm/05710231

**Mathematical Reviews number (MathSciNet)**

MR2648262

**Zentralblatt MATH identifier**

1202.28010

**Subjects**

Primary: 28A80: Fractals [See also 37Fxx] 31C25: Dirichlet spaces 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

**Keywords**

Generalized Sierpinski carpet non-diagonality volume doubling property self-similar measure

#### Citation

Kajino, Naotaka. Remarks on non-diagonality conditions for Sierpinski carpets. Probabilistic Approach to Geometry, 231--241, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710231. https://projecteuclid.org/euclid.aspm/1543086321