A computable absolutely normal Liouville number

AbstractWe consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value 0≤β≤1.

Download Full-text

Introduction to Liouville Numbers

Formalized Mathematics ◽

10.1515/forma-2017-0003 ◽

2017 ◽

Vol 25 (1) ◽

pp. 39-48

Author(s):

Adam Grabowski ◽

Artur Korniłowicz

Keyword(s):

Positive Integer ◽

Algebraic Number ◽

Finite Sequence ◽

Rational Numbers ◽

Liouville Number ◽

Liouville Numbers ◽

Constant Sequence ◽

Transcendental Numbers ◽

Finite Sequences ◽

Definition Of

Summary The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum for a finite sequence {ak}k∈ℕ and b ∈ ℕ. Based on this definition, we also introduced the so-called Liouville number as substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https://en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.

Download Full-text

An "Oceans of Zeros" Proof That a Certain Non-Liouville Number is Transcendental

American Mathematical Monthly ◽

10.2307/2324154 ◽

1991 ◽

Vol 98 (10) ◽

pp. 947 ◽

Cited By ~ 3

Author(s):

M. J. Knight

Keyword(s):

Liouville Number

Download Full-text

All Liouville Numbers are Transcendental

Formalized Mathematics ◽

10.1515/forma-2017-0004 ◽

2017 ◽

Vol 25 (1) ◽

pp. 49-54

Author(s):

Artur Korniłowicz ◽

Adam Naumowicz ◽

Adam Grabowski

Keyword(s):

Positive Integer ◽

Diophantine Approximation ◽

Algebraic Number ◽

Rational Numbers ◽

Algebraic Numbers ◽

Liouville Number ◽

Liouville Numbers ◽

Transcendental Numbers ◽

Algebraic Domains ◽

Liouville’S Theorem

Summary In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.

Download Full-text

Destruction of invariant circles

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700009421 ◽

1988 ◽

Vol 8 (8) ◽

pp. 199-214 ◽

Cited By ~ 35

Keyword(s):

Rotation Number ◽

Invariant Circle ◽

Liouville Number ◽

Invariant Circles ◽

Area Preserving

AbstractLet in be an exact, area-preserving, monotone twist diffeomorphism of the cylinder and let 10 be a Liouville number. We will show that arbitrarily close to in the C∞ topology there exists a C∞ diffeomorphism with no homotopically non-trivial invariant circle of rotation number ω.

Download Full-text