scholarly journals Non-uniform ergodic properties of Hamiltonian flows with impacts

2021 ◽  
pp. 1-63
Author(s):  
KRZYSZTOF FRĄCZEK ◽  
VERED ROM-KEDAR
Keyword(s):  
Energy Level ◽  
Configuration Space ◽  
Level Sets ◽  
Ergodic Averages ◽  
Weighted Averages ◽  
Hamiltonian Flows ◽  
Ergodic Properties ◽  
Topology Changes ◽  
Almost All ◽  
Uniquely Ergodic

Abstract The ergodic properties of two uncoupled oscillators, one horizontal and one vertical, residing in a class of non-rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly greater than $1$ . When at least one of the oscillators is unharmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is uniquely ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components coexist. We prove that for almost all partial energies in such segments the motion is uniquely ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

scholarly journals Pointwise multiple averages for sublinear functions

2018 ◽  
Vol 40 (6) ◽  
pp. 1594-1618
Author(s):  
SEBASTIÁN DONOSO ◽  
ANDREAS KOUTSOGIANNIS ◽  
WENBO SUN
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
Pointwise Convergence ◽  
Limit Function ◽  
Invariant Property ◽  
Ergodic Averages ◽  
Order Zero ◽  
Good For ◽  
Sublinear Functions ◽  
Uniquely Ergodic

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

scholarly journals Level sets of multiple ergodic averages

2011 ◽  
Vol 168 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Ai-Hua Fan ◽  
Lingmin Liao ◽  
Ji-Hua Ma
Keyword(s):  
Level Sets ◽  
Ergodic Averages

Labor Force Transitions by Gender: Implications for Separate and Combined Worklife Expectancy

2020 ◽  
Author(s):  
Craig A. Allen
Keyword(s):  
Labor Force ◽  
Labor Force Participation ◽  
Total Population ◽  
Transition Probabilities ◽  
Participation Rates ◽  
Weighted Averages ◽  
Almost All ◽  
Taking Care

Abstract The Fair Calculations in Civil Damages Act of 2016 (the “Act”) proposes that tables of worklife expectancy not taking account of gender be used in the calculation of damages amounts for loss of earnings capacity. This study calculates worklife expectancy tables not taking account of gender, by taking weighted averages of the labor force transition probabilities used by Skoog-Ciecka-Krueger (2011) and calculating the resulting worklife expectancies using the method of Skoog (2002). The weights for the weighted averages are from the period January 2005 through December 2009. This study then examines patterns in worklife expectancies and labor force transition probabilities and provides evidence not in support of the hypothesis that women's lower worklife expectancies are due to events limited to childbearing years. It is seen that at almost all ages and at all levels of education: women's rate of departure from the labor force exceeds that of men, women's rate of entry and return to the labor force is less than that of men, and labor force participation rates are less than those of men. This study finds that women's lower labor force participation rates than those of men are accounted for by a greater incidence for women of the statuses “taking care of house or family” and “in retirement.” Combined gender worklife expectancy tables are then applied to the total population, with the result that mean worklife expectancies are higher for women than men at ages 30 and under, and progressively lower for women than men as age increases beyond age 30.

Measurements Of The Hydrogen Absorption Pressurecomposition Isotherms IN Ti/Zr-Based Quasicrystals

1998 ◽  
Vol 553 ◽  
Author(s):  
J. Y. Kim ◽  
E. H. Majzoub ◽  
P. C. Gibbons ◽  
K. F. Kelton
Keyword(s):  
Energy Distribution ◽  
Hydrogen Absorption ◽  
Structural Similarity ◽  
Half Width ◽  
Weighted Averages ◽  
Site Energy ◽  
Tetrahedral Sites ◽  
Two Phases ◽  
Almost All ◽  
Broad Energy

AbstractThe first hydrogen absorption pressure-composition isotherms (p-c-T) were measured in quasicrystalline Ti45Zr38Ni17. No evidence for a pressure plateau was found, indicating a distribution of energies for the hydrogen in interstitial sites. Fits to the p-c-T data confirmed this, giving energy peaks at -0.19 eV with a full width at half maximum (half-width) of 0.06 eV, and at -0.09 eV with a half-width of 0.08 eV. This is in contrast with the broad site energy distribution that is characteristic of a metallic glass. In agreement, fits to data taken from amorphous Ti45Zr27Ni20Si8 gave a single broad energy distribution at -0.10 eV with a half-width of 0.35 eV. Based on the weighted averages of the site energies for the pure components, the energies assigned to the tetrahedral sites in the Ti44Zr40Ni16 1/1 approximant phase are in qualitative agreement with the measured data for the quasicrystal, supporting a local structural similarity between these two phases. Almost all of the absorbed hydrogen can be desorbed at 650°C in one hour by pumping, without transforming the quasicrystal phase and without powdering the rapidly-quenched samples.

scholarly journals Simplicial systems for interval exchange maps and measured foliations

1985 ◽  
Vol 5 (2) ◽  
pp. 257-271 ◽  
Author(s):  
S. P. Kerckhoff
Keyword(s):  
General Theorem ◽  
Interval Exchange ◽  
Alternative Proof ◽  
Simple Assumption ◽  
Refinement Process ◽  
Almost All ◽  
Interval Exchange Maps ◽  
Uniquely Ergodic

AbstractThe spaces of interval exchange maps and measured foliations are considered and an alternative proof that almost all interval exchange maps and measured foliations are uniquely ergodic is given. These spaces are endowed with a refinement process, called a simplicial system, which is studied abstractly and is shown to be normal under a simple assumption. The results follow and thus are a corollary of a more general theorem in a broader setting.

scholarly journals Equidistribution of singular measures on nilmanifolds and skew products

2011 ◽  
Vol 31 (6) ◽  
pp. 1785-1817
Author(s):  
FABRIZIO POLO
Keyword(s):  
Haar Measure ◽  
Uniform Density ◽  
Unipotent Group ◽  
Affine Transformations ◽  
Skew Products ◽  
Singular Measures ◽  
Ergodic Properties ◽  
Push Forward ◽  
Multiplicative Ergodic Theorem ◽  
Uniquely Ergodic

AbstractWe prove that for a minimal rotationTon a two-step nilmanifold and any measureμ, the push-forwardTn⋆μofμunderTntends toward Haar measure if and only ifμprojects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds.Amer. J. Math.91(1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus.Amer. J. Math.83(1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits ofTn⋆μfor some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

scholarly journals An alternative approach to the ergodic theory of measured foliations on surfaces

1981 ◽  
Vol 1 (4) ◽  
pp. 461-488 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Projective Space ◽  
Lebesgue Measure ◽  
Ergodic Theory ◽  
Diffeomorphism Group ◽  
Alternative Approach ◽  
Interval Exchanges ◽  
Almost All ◽  
Uniquely Ergodic

AbstractWe consider measured foliations on surfaces, and interval exchanges. We give alternative proofs of the following theorems first proved by Masur and (independently) Veech. The action of the diffeomorphism group of the surface on the projective space of measured foliations (with respect to a natural ‘Lebesgue’ measure) is ergodic. Almost all measured foliations are uniquely ergodic. Almost all interval exchanges (again, with respect to a natural ‘Lebesgue’ measure) are uniquely ergodic.

Deviation for interval exchange transformations

1997 ◽  
Vol 17 (6) ◽  
pp. 1477-1499 ◽  
Author(s):  
ANTON ZORICH
Keyword(s):  
Ergodic Theorem ◽  
Gauss Map ◽  
Interval Exchange ◽  
Actual Number ◽  
Interval Exchange Transformation ◽  
Exchange Transformation ◽  
Interval Exchange Transformations ◽  
Starting Point ◽  
Almost All ◽  
Uniquely Ergodic

Consider a long piece of a trajectory $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$ of an interval exchange transformation $T$. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subinterval would be approximately $\lambda_i n$. Here $\lambda_i$ is the length of the corresponding subinterval of our unit interval $X$. In this paper we give an estimate for the deviation of the actual number of visits to the $i$th subinterval $X_i$ from one predicted by the ergodic theorem.We prove that for almost all interval exchange transformations the following bound is valid: $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} < 1. $$ Roughly speaking the error term is bounded by $n^{\theta_2/\theta_1}$. The numbers $0\le \theta_2 < \theta_1$ depend only on the permutation $\pi$ corresponding to the interval exchange transformation (actually, only on the Rauzy class of the permutation). In the case of interval exchange of two intervals we obviously have $\theta_2=0$. In the case of exchange of three and more intervals the numbers $\theta_1, \theta_2$ are the two top Lyapunov exponents related to the corresponding generalized Gauss map on the space of interval exchange transformations.The limit above ‘converges to the bound’ uniformly for all $x\in X$ in the following sense. For any $\varepsilon >0$ the ratio of logarithms would be less than $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $ for all $n\ge N(\varepsilon)$, where $N(\varepsilon)$ does not depend on the starting point $x\in X$.

“Hot spots” in high-latitude moss-associated N fixation: What drives locally high fixation rates?

2020 ◽  
Author(s):  
Julia Stuart ◽  
Michelle Mack
Keyword(s):  
Hot Spots ◽  
Large Scale ◽  
Mixed Model ◽  
Standard Errors ◽  
N Fixation ◽  
Taxonomic Richness ◽  
Moss Species ◽  
Weighted Averages ◽  
The Mean ◽  
Almost All

&lt;p&gt;Moss-associated nitrogen (N) fixation provides a substantial but heterogeneous input of new N to nutrient limited ecosystems at high latitudes. The presence of &amp;#8220;hot spots&amp;#8221;, defined as a rate of N fixation greater than three standard errors over the mean rate, can further increase the difficulty of scaling N inputs to plant communities or ecosystems. We used &lt;sup&gt;15&lt;/sup&gt;N&lt;sub&gt;2&lt;/sub&gt; incubations to quantify the fixation rates associated with 34 moss species from 24 sites ranging from 60 to 68 degrees N in Alaska, USA. The total moss-associated fixation rates ranged from 0.08 to 4.4 kg N ha&lt;sup&gt;-1&lt;/sup&gt;yr&lt;sup&gt;-1&lt;/sup&gt;, with an average of 1.1 kg N ha&lt;sup&gt;-1&lt;/sup&gt;yr&lt;sup&gt;-1&lt;/sup&gt;, based on abundance-weighted averages of all mosses summed for each site. Five of the 24 sampled sites were hot spots of N fixation. We hypothesized that host moss diversity would be correlated with higher N fixation rates, since different mosses often have distinct microbial assemblages and higher microbial diversity has been linked with higher N fixation rates in other ecosystems. However, we found no significant correlation between either moss taxonomic richness or Simpson&amp;#8217;s D and N fixation rates (p=0.102, R&lt;sup&gt;2&lt;/sup&gt;=0.01 and p=0.522, R&lt;sup&gt;2&lt;/sup&gt;=0.02, respectively). What we found instead was that certain high-fixing species, most importantly &lt;em&gt;Tomentypnum nitens&lt;/em&gt;, were present in almost all hot spots. The relevance of moss taxonomic identity in driving N fixation rates was repeatedly observed in our survey, where both machine learning and mixed model approaches found that moss family was a significant predictor of associated fixation rates across ecosystems in Alaska. Taken together, these results indicate the importance of moss identity in driving hot spots and illustrate that host taxonomy may be a useful tool in generating more accurate large-scale assessments of associated N inputs in these vulnerable and valuable ecosystems.&lt;/p&gt;

Sign in / Sign up

Export Citation Format

Share Document