Problem of Erdös – Vershik for Golden Ratio
نویسنده
چکیده
منابع مشابه
Ergodic Properties of Erdös Measure , the Entropy of the Goldenshift , and Related Problems
We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on T which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism...
متن کامل2 7 O ct 1 99 8 ERGODIC PROPERTIES OF ERDÖS MEASURE , THE ENTROPY OF THE GOLDENSHIFT , AND RELATED PROBLEMS
We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on T which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism...
متن کاملSimilar Triangles, Another Trace of the Golden Ratio
In this paper we investigate similar triangles which are not congruent but have two pair congruent sides. We show that greatest lower bound of values for similarity ratio of such triangles is golden ratio. For right triangles, we prove that the supremum of values for similarity ratio is the square root of the golden ratio.
متن کاملErgodic-theoretic Properties of Certain Bernoulli Convolutions
In [17] the author and A. Vershik have shown that for β = 1 2 (1 + √ 5) and the alphabet {0, 1} the infinite Bernoulli convolution (= the Erdös measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type II1 under the β-shift, and the natural extension of the β-shift provided with the measure equivalent to the Erdös measure, is Bernoulli. In this note we exten...
متن کاملA Class of Convergent Series with Golden Ratio Based on Fibonacci Sequence
In this article, a class of convergent series based on Fibonacci sequence is introduced for which there is a golden ratio (i.e. $frac{1+sqrt 5}{2}),$ with respect to convergence analysis. A class of sequences are at first built using two consecutive numbers of Fibonacci sequence and, therefore, new sequences have been used in order to introduce a new class of series. All properties of the se...
متن کامل