## Rocky Mountain Journal of Mathematics

- Rocky Mountain J. Math.
- Volume 48, Number 4 (2018), 1359-1384.

### Smale spaces from self-similar graph actions

#### Abstract

We show that, for contracting and regular self-similar graph actions, the shift maps on limit spaces are positively expansive local homeomorphisms. From this, we find that limit solenoids of contracting and regular self-similar graph actions are Smale spaces and that the unstable Ruelle algebras of the limit solenoids are strongly Morita equivalent to the Cuntz-Pimsner algebras by Exel and Pardo if self-similar graph actions satisfy the contracting, regular, pseudo free and $G$-transitive conditions.

#### Article information

**Source**

Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1359-1384.

**Dates**

First available in Project Euclid: 30 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.rmjm/1538272837

**Digital Object Identifier**

doi:10.1216/RMJ-2018-48-4-1359

**Mathematical Reviews number (MathSciNet)**

MR3859762

**Zentralblatt MATH identifier**

06958783

**Subjects**

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 46L05: General theory of $C^*$-algebras 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

**Keywords**

Self-similar graph action limit dynamical system limit solenoid positively expansive map Smale space

#### Citation

Yi, Inhyeop. Smale spaces from self-similar graph actions. Rocky Mountain J. Math. 48 (2018), no. 4, 1359--1384. doi:10.1216/RMJ-2018-48-4-1359. https://projecteuclid.org/euclid.rmjm/1538272837