scholarly journals Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms

2008 ◽  
Vol 237 (20) ◽  
pp. 2599-2615 ◽  
Author(s):  
Alejandro Luque ◽  
Jordi Villanueva
Keyword(s):  
Rotation Number ◽  
Parametric Families ◽  
Circle Diffeomorphisms ◽  
Derivatives Of

scholarly journals Dimensional characteristics of invariant measures for circle diffeomorphisms

2009 ◽  
Vol 29 (6) ◽  
pp. 1979-1992 ◽  
Author(s):  
VICTORIA SADOVSKAYA
Keyword(s):  
Invariant Measure ◽  
Rotation Number ◽  
Invariant Measures ◽  
Liouville Number ◽  
Hausdorff Dimensions ◽  
Rotation Numbers ◽  
Circle Diffeomorphism ◽  
Unique Invariant Measure ◽  
Circle Diffeomorphisms ◽  
Box Dimensions

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.

Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point

2017 ◽  
Vol 39 (5) ◽  
pp. 1331-1339
Author(s):  
KONSTANTIN KHANIN ◽  
SAŠA KOCIĆ
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Rotation Number ◽  
Break Point ◽  
Irrational Rotation ◽  
Rotation Numbers ◽  
Circle Diffeomorphism ◽  
Almost All ◽  
Circle Diffeomorphisms

We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$, the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$-smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$, with rotation number $\unicode[STIX]{x1D70C}$, is zero. This result cannot be extended to all irrational rotation numbers.

Rotation number and one-parameter families of circle diffeomorphisms

1992 ◽  
Vol 12 (2) ◽  
pp. 359-363 ◽  
Author(s):  
Masato Tsujii
Keyword(s):  
Periodic Orbits ◽  
Rotation Number ◽  
Rational Number ◽  
Irrational Number ◽  
Rigid Rotation ◽  
Circle Diffeomorphisms

AbstractWe consider one-parameter families of circle diffeomorphisms, f1(x) = f(x) + t(t ∈ S1), where f: S1 is a Cr-diffeomorphism (r≥3). We show that, for Lebesgue almost every t ∈ S1 the rotation number of f1, is either a rational number or an irrational number of Roth type. In the former case, f1, has periodic orbits and, in the latter case, f1, is Cr − 2-conjugate to an irrational rigid rotation from well-known theorems of Herman and Yoccoz.

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