Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multiplicity structure of preimages of invariant measures under finite-to-one factor maps


Author: Jisang Yoo
Journal: Trans. Amer. Math. Soc. 370 (2018), 8111-8133
MSC (2010): Primary 37B10; Secondary 37A99, 37B15
DOI: https://doi.org/10.1090/tran/7234
Published electronically: May 3, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a finite-to-one factor map $ \pi : (X, T) \to (Y, S)$ between topological dynamical systems, we look into the pushforward map $ \pi _*: M(X, T) \to M(Y,S)$ between sets of invariant measures. We investigate the structure of the measure fiber $ \pi _*^{-1}(\nu )$ for an arbitrary ergodic measure $ \nu $ on the factor system $ Y$. We define the degree $ d_{\pi ,\nu }$ of the factor map $ \pi $ relative to $ \nu $ and the multiplicity of each ergodic measure $ \mu $ on $ X$ that projects to $ \nu $, and show that the number of ergodic preimages of $ \nu $ is $ d_{\pi ,\nu }$ counting multiplicity. In other words, the degree $ d_{\pi ,\nu }$ is the sum of the multiplicity of $ \mu $, where $ \mu $ runs over the ergodic measures in the measure fiber $ \pi ^{-1}_*(\nu )$. This generalizes the following folklore result in symbolic dynamics for lifting fully supported invariant measures: Given a finite-to-one factor code $ \pi : X \to Y$ between irreducible sofic shifts and an ergodic measure $ \nu $ on $ Y$ with full support, $ \pi ^{-1}_*(\nu )$ has at most $ d_\pi $ ergodic measures in it, where $ d_\pi $ is the degree of $ \pi $. We apply our theory of the structure of measure fibers to the special case of symbolic dynamical systems. In this case, we demonstrate that one can list all (finitely many) ergodic measures in the measure fiber $ \pi ^{-1}_*(\nu )$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37B10, 37A99, 37B15

Retrieve articles in all journals with MSC (2010): 37B10, 37A99, 37B15


Additional Information

Jisang Yoo
Affiliation: Seoul National University, Seoul, South Korea
Address at time of publication: Sungkyunkwan University, Suwon, South Korea
Email: jisangy@kaist.ac.kr

DOI: https://doi.org/10.1090/tran/7234
Keywords: Degree, factor code, SFT, sofic subshfit, factor map, finite-to-one, invariant measure
Received by editor(s): December 27, 2016
Received by editor(s) in revised form: March 6, 2017
Published electronically: May 3, 2018
Additional Notes: This research was supported by BK21 PLUS SNU Mathematical Sciences Division. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A6A3A01040839) and the National Research Foundation of Korea (NRF) grant funded by the MEST 2015R1A3A2031159.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society