Book contents
- Frontmatter
- Contents
- Introduction
- Surveys
- Research Papers
- 1 Uniformity in the polynomial Szemérdi theorem
- 2 Some 2-d symbolic dynamical systems: Entropy and mixing
- 3 A note on certain rigid subshifts
- 4 Entropy of graphs, semigroups and groups
- 5 On representation of integers in Linear Numeration Systems
- 6 The structure of ergodic transformations conjugate to their inverses
- 7 Approximatiom by periodic transformations and diophantine approximation of the spectrum
- 8 Invariant σ-algebras for ℤd-actions and their applications
- 9 Large deviations for paths and configurations counting
- 10 A zeta function for ℤd-actions
- 11 The dynamical theory of tilings and Quasicrystallography
4 - Entropy of graphs, semigroups and groups
Published online by Cambridge University Press: 30 March 2010
- Frontmatter
- Contents
- Introduction
- Surveys
- Research Papers
- 1 Uniformity in the polynomial Szemérdi theorem
- 2 Some 2-d symbolic dynamical systems: Entropy and mixing
- 3 A note on certain rigid subshifts
- 4 Entropy of graphs, semigroups and groups
- 5 On representation of integers in Linear Numeration Systems
- 6 The structure of ergodic transformations conjugate to their inverses
- 7 Approximatiom by periodic transformations and diophantine approximation of the spectrum
- 8 Invariant σ-algebras for ℤd-actions and their applications
- 9 Large deviations for paths and configurations counting
- 10 A zeta function for ℤd-actions
- 11 The dynamical theory of tilings and Quasicrystallography
Summary
Introduction
Let X be a compact metric space and T : X → X is continuous transformation. Then the dynamics of T is a widely studied subject. In particular, h(T) – the entropy of T is a well understood object. Let г ⊂ X × X be a closed set. Then г induces certain dynamics and entropy h(г). If X is a finite set then г can be naturally viewed as a directed graph. That is, if X = {1, …, n} then г consists of all directed arcs i → j so that (i, j) ∈ г. Then г induces a subshift of finite type which is a widely studied subject. However, in the case that X is infinite, the subject of dynamic of г and its entropy are relatively new. The first paper treating the entropy of a graph is due to [Gro]. In that context X is a compact Riemannian manifold and г can be viewed as a Riemannian submanifold. (Actually, г can have singularities.) We treated this subject in [Fril-3]. See Bullet [Bul1-2] for the dynamics of quadratic correspondences and [M-R] for iterated algebraic functions.
The object of this paper is to study the entropy of a corresponding map induced by г. We now describe briefly the main results of the paper. Let X be a compact metric space and assume that г ⊂ X × X is a closed set.
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- Ergodic Theory and Zd Actions , pp. 319 - 344Publisher: Cambridge University PressPrint publication year: 1996
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