scholarly journals Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Stefano Galatolo ◽  
Alfonso Sorrentino
Keyword(s):  
Rotation Number ◽  
Linear Response ◽  
Transfer Operator ◽  
Kam Theory ◽  
Hölder Exponent ◽  
Statistical Stability ◽  
Rotation Numbers ◽  
Circle Diffeomorphisms ◽  
Stability Results ◽  
Spaces Of Measures

<p style='text-indent:20px;'>We prove quantitative statistical stability results for a large class of small <inline-formula><tex-math id="M1">\begin{document}$ C^{0} $\end{document}</tex-math></inline-formula> perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.</p>

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.

scholarly journals Statistical stability and linear response for random hyperbolic dynamics

2021 ◽  
pp. 1-30
Author(s):  
DAVOR DRAGIČEVIĆ ◽  
JULIEN SEDRO
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Linear Response ◽  
Ad Hoc ◽  
Transfer Operator ◽  
Statistical Stability ◽  
Hyperbolic Dynamics ◽  
Quenched Central Limit Theorem ◽  
Random Products

Abstract We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.

Influence of Radial Rotation on Heat Transfer in a Rectangular Channel With Two Opposite Walls Roughened by Hemispherical Protrusions at High Rotation Numbers

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Shyy Woei Chang ◽  
Tong-Miin. Liou ◽  
Wei-Chun Chen
Keyword(s):  
Heat Transfer ◽  
Infrared Thermography ◽  
Rotation Number ◽  
Rectangular Channel ◽  
Flow Region ◽  
Transfer Data ◽  
Nusselt Numbers ◽  
Rotation Numbers ◽  
Heat Transfer Correlations ◽  
Rotating Channel

Detailed heat transfer distributions over two opposite leading and trailing walls roughened by hemispherical protrusions were measured from a rotating rectangular channel at rotation number up to 0.6 to examine the effects of Reynolds (Re), rotation (Ro), and buoyancy (Bu) numbers on local and area-averaged Nusselt numbers (Nu and Nu¯) using the infrared thermography. A set of selected heat transfer data illustrates the Coriolis and rotating buoyancy effects on the detailed Nu distributions and the area-averaged heat transfer performances of the rotating channel. The Nu¯ for the developed flow region on the leading and trailing walls are parametrically analyzed to devise the empirical heat transfer correlations that permit the evaluation of the interdependent and individual Re, Ro, and Bu effect on Nu¯.

Knotted periodic orbits in suspensions of annulus maps

1987 ◽  
Vol 411 (1841) ◽  
pp. 351-378 ◽  
Keyword(s):  
Periodic Orbits ◽  
Rotation Number ◽  
Existence And Uniqueness ◽  
Torus Knots ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Rotation Numbers ◽  
Higher Genus

We consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3-manifold T 2 x I. Using results of Birman & Williams ( Topology 22, 47‒82 (1983); Contemp. Math . 20, 1‒60 (1983)), we construct a knotholder or template that carries the set of periodic orbits of the flow. We define rotation numbers and show that any orbit of period q and rotation number p / q can be arranged as a positive braid on p strands. This yields existence and uniqueness results for families of resonant torus knots ( p -braids that are ( p , q )-torus knots of period q > p which correspond to order-preserving (Birkhoff-) periodic orbits of the diffeomorphism. We show that all other q -periodic p -braids have higher genus, and we establish bounds on the genera of such knots. We obtain existence and uniqueness results for a number of other, non-resonant, torus knots, including non-order-preserving ( q + s , q )-torus knots of rotation number 1.

Direct numerical simulations of rotating turbulent channel flow

2008 ◽  
Vol 598 ◽  
pp. 177-199 ◽  
Author(s):  
OLOF GRUNDESTAM ◽  
STEFAN WALLIN ◽  
ARNE V. JOHANSSON
Keyword(s):  
Numerical Simulations ◽  
Channel Flow ◽  
Rotation Number ◽  
Turbulent Channel Flow ◽  
Shear Velocity ◽  
Direct Numerical Simulations ◽  
A Value ◽  
Rotation Numbers ◽  
Wall Shear ◽  
Two Sides

Fully developed rotating turbulent channel flow has been studied, through direct numerical simulations, for the complete range of rotation numbers for which the flow is turbulent. The present investigation suggests that complete flow laminarization occurs at a rotation number Ro = 2Ωδ/Ub ≤ 3.0, where Ω denotes the system rotation, Ub is the mean bulk velocity and δ is the half-width of the channel. Simulations were performed for ten different rotation numbers in the range 0.98 to 2.49 and complemented with earlier simulations (done in our group) for lower values of Ro. The friction Reynolds number Reτ = uτδ/ν (where uτ is the wall-shear velocity and ν is the kinematic viscosity) was chosen as 180 for these simulations. A striking feature of rotating channel flow is the division into a turbulent (unstable) and an almost laminarized (stable) side. The relatively distinct interface between these two regions was found to be maintained by a balance where negative turbulence production plays an important role. The maximum difference in wall-shear stress between the two sides was found to occur for a rotation number of about 0.5. The bulk flow was found to monotonically increase with increasing rotation number and reach a value (for Reτ = 180) at the laminar limit (Ro = 3.0) four times that of the non-rotating case.

Effect of Rib Size on Heat (Mass) Transfer Distribution in a Rotation Channel

1997 ◽  
Author(s):  
C. W. Park ◽  
R. T. Kukreja ◽  
S. C. Lau
Keyword(s):  
Mass Transfer ◽  
Rotation Number ◽  
Heat Mass Transfer ◽  
Reynolds Numbers ◽  
Local Heat ◽  
Test Channel ◽  
Turbine Blade Cooling ◽  
Blade Cooling ◽  
Rotation Numbers ◽  
Better Than

Experiments have been conducted to study the effect of rib size on the local heat (mass) transfer distribution for radial outward flow in a rotating channel with transverse ribs on the leading and trailing walls. The test channel modeled internal turbine blade cooling passages. Results were obtained for Reynolds numbers of 5,500 and 10,000, rotation numbers of 0.09 and 0.24, and for a fixed rib pitch that was equal to the channel hydraulic diameter. For a fixed rib configuration on the leading wall, increasing the size of the ribs on the trailing wall increased the heat (mass) transfer on the leading wall. Ribs with D/e = p/e = 16 on the trailing wall performed better than ribs with D/e = p/e = 10. When the rotation number was large, the heat (mass) transfer on the leading wall was quite low, regardless of the sizes of the ribs on the leading and trailing walls. There was very little spanwise variation of the local heat (mass) transfer between the transverse ribs on the trailing wall. When the rotation number was large, however, there was a significant spanwise variation of the local heat (mass) transfer between ribs on the leading wall.

OBSERVED ROTATION NUMBERS IN FAMILIES OF CIRCLE MAPS

2001 ◽  
Vol 11 (01) ◽  
pp. 73-89 ◽  
Author(s):  
MICHAEL A. SAUM ◽  
TODD R. YOUNG
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Narrow Band ◽  
Initial Conditions ◽  
Positive Probability ◽  
Circle Maps ◽  
Numerical Evidence ◽  
Rotation Interval ◽  
Large Numbers ◽  
Rotation Numbers

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers.

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