Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation

1967 ◽  
Vol 1 (1) ◽  
pp. 1-49 ◽  
Author(s):  
Harry Furstenberg
Keyword(s):  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Minimal Sets

scholarly journals Symbolic dynamics in mean dimension theory

2020 ◽  
pp. 1-19
Author(s):  
MAO SHINODA ◽  
MASAKI TSUKAMOTO
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Symbolic Dynamics ◽  
Rate Distortion ◽  
Dimension Theory ◽  
Minimal Sets ◽  
Mean Dimension ◽  
The Mean

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^{2}$ -subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of $\mathbb{Z}^{2}$ -subshifts with respect to a subaction of $\mathbb{Z}$ . The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^{2}$ -subshifts in terms of Kolmogorov–Sinai entropy.

Lattice orbit distribution on ℝ2

2009 ◽  
Vol 30 (4) ◽  
pp. 1201-1214 ◽  
Author(s):  
ARNALDO NOGUEIRA
Keyword(s):  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Error Term ◽  
Absolute Error ◽  
Compact Set ◽  
Linear Action ◽  
Counting Problem ◽  
Asymptotical Behaviour ◽  
Diophantine Property ◽  
The Absolute

AbstractWe study the distribution on ℝ2 of the orbit of a vector under the linear action of SL(2,ℤ). Let Ω⊂ℝ2 be a compact set and x∈ℝ2. Let N(k,x) be the number of matrices γ∈SL(2,ℤ) such that γ(x)∈Ω and ‖γ‖≤k, k=1,2,…. If Ω is a square, we prove the existence of an absolute error term for N(k,x), as k→∞, for almost every x, which depends on the Diophantine property of the ratio of the coordinates of x. Our approach translates the question into a Diophantine approximation counting problem which provides the absolute error term. The asymptotical behaviour of N(k,x) is also obtained using ergodic theory.

scholarly journals Ergodic theory and Diophantine approximation for translation surfaces and linear forms

Nonlinearity ◽  
2016 ◽  
Vol 29 (8) ◽  
pp. 2173-2190 ◽  
Author(s):  
Jayadev Athreya ◽  
Andrew Parrish ◽  
Jimmy Tseng
Keyword(s):  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Translation Surfaces ◽  
Linear Forms

Ergodic Theory

1983 ◽  
Author(s):  
Karl E. Petersen
Keyword(s):  
Ergodic Theory

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

Sign in / Sign up

Export Citation Format

Share Document