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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Statistical stability for multi-substitution tiling spaces

Pages: 4579 - 4594, Volume 33, Issue 10, October 2013      doi:10.3934/dcds.2013.33.4579

 
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Rui Pacheco - Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, Covilhã, 6200-001, Portugal (email)
Helder Vilarinho - Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, Covilhã, 6200-001, Portugal (email)

Abstract: Given a finite set $\{S_1\dots,S_k \}$ of substitution maps acting on a certain finite number (up to translations) of tiles in $\mathbb{R}^d$, we consider the multi-substitution tiling space associated to each sequence $\bar a\in \{1,\ldots,k\}^{\mathbb{N}}$. The action by translations on such spaces gives rise to uniquely ergodic dynamical systems. In this paper we investigate the rate of convergence for ergodic limits of patches frequencies and prove that these limits vary continuously with $\bar a$.

Keywords:  Multi-substitutions, tiling spaces, dynamical systems, invariant measures, statistical stability.
Mathematics Subject Classification:  Primary: 37A15, 37A25, 52C22.

Received: July 2012;      Revised: January 2013;      Available Online: April 2013.

 References