Quantitative behavior of non-integrable systems. II
- PDF / 3,710,025 Bytes
- 105 Pages / 476.22 x 680.315 pts Page_size
- 35 Downloads / 191 Views
DOI: 0
QUANTITATIVE BEHAVIOR OF NON-INTEGRABLE SYSTEMS. II J. BECK∗ , M. DONDERS and Y. YANG Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, Piscataway NJ 08854, USA e-mails: [email protected], [email protected], [email protected] (Received October 6, 2019; accepted January 15, 2020)
Abstract. Here we finish the proof of the Main Theorem on the L-surface, i.e., Theorem 2.1.4 in part (I) of this paper [1], to which the reader is also referred for basic notation. In Sections 3–5 we develop the basic form of the shortline method. In the subsequent papers this basic form will be be extended and modified in many different ways.
4. L-surface: more irregularity exponents 4.1. L-surface: case of the shortest period. First we summarize the result of Section 3 in our earlier paper [1] in a nutshell. Much of Section 3 is concerned with the special √ L-lines V0 (t) and H0 (t), starting from √ the origin 0 and having slopes α = 2 + 1 and its reciprocal 1/α = β = 2 − 1. By using the mutual shortline property of these special L-lines, we prove Theorem 3.2.3, the main result of the chapter. It precisely describes the quantitative aspects of equidistribution for any L-line of slope α or β relative to all convex subsets (a remarkably large family of test sets). It gives the exact order of the fluctuations around perfect triple uniformity, namely the polynomial order T κ0 , where √
log 1+2 κ0 = log α
5
√
log 1+2 5 √ . = log(1 + 2)
The corresponding continued fraction expansions 1+
√ 2=2+
1 2+
1 2+···
= [2; 2, 2, 2, . . .]
∗ Corresponding
author. Key words and phrases: flows on surfaces, uniformity of sequences and sets. Mathematics Subject Classification: 37E35, 11B50, 11B30.
c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
22
J. J. BECK, BECK, M. M. DONDERS DONDERS and and Y. Y. YANG YANG
and
√ 1 1+ 5 =1+ = [1; 1, 1, 1, . . .] 1 2 1 + 1+···
exhibit the simple rule of “digit halving”. We call κ0 the irregularity exponent of slopes α and β. Note that fluctuations of size T κ0 really show up infinitely many times even for such a simple test set as a whole square face; see Theorem 3.2.3. We prove superdensity; see Theorem 3.4.1. We also prove discrete uniformity of almost all arithmetic progressions on any L-line having the slope α or β; see Theorem 2.2.3. √ The “new number” (1 + 5)/2 in the definition of the irregularity exponent κ0 comes from the transition matrix of the shortline method. More precisely, we have the transition matrix M1 using the Delete-End Rule (see (3.1.19)), and also the transition matrix M2 using the Keep-End Rule (see (3.1.21)). These two matrices are “almost the same” in the sense that they have precisely the same eigenvalues √ √ √ −1 ± 3i 1+ 5 , λ3,4 = , λ1 = 1 + 2 = α, λ2 = − 2 2 √ √ 5−1 , λ6 = 1 − 2. λ5 = 2 They also have precisely the same eigenvectors corresponding to the two relevant eigenvalues λ1 and λ2 . (Since we use arbitrarily large powers of the ei
Data Loading...
