# Moscow Mathematical Journal

Volume 19, Issue 1, January–March 2019 pp. 77–88.

Random Averaging in Ergodic Theorem and Boundary Deformation Rate in Symbolic Dynamics

**Authors**:
B.M. Gurevich (1) (2), S.A. Komech (2), and A.A. Tempelman (3)

**Author institution:**(1) Dept. Mech. and Math. Moscow State University, 119991 GSP-1, Moscow, Russia

(2) IITP RAS, Bolshoy Karetny per. 19, build. 1, Lab. 4, Moscow 127051 Russia

(3) Penn State University, University Park, PA 16802, USA

**Summary: **

For some symbolic dynamical systems we study the value of the boundary deformation for a small ball in the phase space during a period of time depending on the center and radius of the ball. For actions of countable Abelian groups, a version of the Mean Ergodic theorem with averaging over random sets is proved and used in the proof of the main theorem on deformation rate.

2010 Math. Subj. Class. 28D20, 37A05, 37A30, 37A50, 37B10.

**Keywords:**Symbolic dynamical systems, topological Markov shift, sofic system, synchronized system, magic word, invariant measure, metric entropy, Mean Ergodic theorem, boundary deformation rate.

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