scholarly journals Minimal systems with finitely many ergodic measures

2021 ◽  
pp. 109000
Author(s):  
Wen Huang ◽  
Zhengxing Lian ◽  
Song Shao ◽  
Xiangdong Ye
Keyword(s):  
Ergodic Measures

scholarly journals DIMENSION THEORY OF LINEAR SOLENOIDS

Fractals ◽  
2007 ◽  
Vol 15 (01) ◽  
pp. 63-72 ◽  
Author(s):  
JÖRG NEUNHÄUSERER
Keyword(s):  
Dimension Theory ◽  
Ergodic Measures ◽  
Full Dimension ◽  
Fractal Attractor

We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question of whether there exists a measure of full dimension.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

Dichotomy of ergodic measures on linear spaces

1995 ◽  
Vol 58 (6) ◽  
pp. 1357-1359 ◽  
Author(s):  
S. G. Khaliullin
Keyword(s):  
Linear Spaces ◽  
Ergodic Measures

scholarly journals ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xingjie Helen Li ◽  
Fei Lu ◽  
Felix X.-F. Ye
Keyword(s):  
Large Time ◽  
Gradient System ◽  
Numerical Schemes ◽  
Time Step ◽  
Inference Problem ◽  
Time Dynamics ◽  
Efficient Simulation ◽  
Time Flow ◽  
Ergodic Measures ◽  
Locally Lipschitz

<p style='text-indent:20px;'>Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.</p><p style='text-indent:20px;'>We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.</p>

Ergodic properties and Weyl M-functions for random linear Hamiltonian systems

2000 ◽  
Vol 130 (5) ◽  
pp. 1045-1079 ◽  
Author(s):  
R. Johnson ◽  
S. Novo ◽  
R. Obaya
Keyword(s):  
Hamiltonian Systems ◽  
Qualitative Description ◽  
Invariant Region ◽  
Absolutely Continuous ◽  
Hamiltonian Equations ◽  
Radial Limits ◽  
Ergodic Properties ◽  
Ergodic Measures ◽  
Linear Hamiltonian Systems ◽  
Dynamical Information

This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.

scholarly journals Dimension of ergodic measures and currents on

2019 ◽  
Vol 40 (8) ◽  
pp. 2131-2155
Author(s):  
CHRISTOPHE DUPONT ◽  
AXEL ROGUE
Keyword(s):  
Upper Bound ◽  
Equilibrium Measure ◽  
Pointwise Dimension ◽  
Ergodic Measures ◽  
Positive Currents ◽  
Green Current

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^{2}$ of degree $d\geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $\unicode[STIX]{x1D708}$. We infer several applications. The first one is an upper bound for the lower pointwise dimension of the equilibrium measure, towards a Binder–DeMarco’s formula for this dimension. The second one shows that every current $S$ containing a measure of entropy $h_{\unicode[STIX]{x1D708}}>\log d$ has a directional dimension ${>}2$, which answers a question of de Thélin–Vigny in a directional way. The last one estimates the dimensions of the Green current of Dujardin’s semi-extremal endomorphisms.

scholarly journals The structure of the space of ergodic measures of transitive partially hyperbolic sets

2019 ◽  
Vol 190 (3) ◽  
pp. 441-479
Author(s):  
L. J. Díaz ◽  
K. Gelfert ◽  
T. Marcarini ◽  
M. Rams
Keyword(s):  
Ergodic Measures ◽  
Partially Hyperbolic ◽  
Hyperbolic Sets

scholarly journals Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

2020 ◽  
Vol 2020 (758) ◽  
pp. 1-66
Author(s):  
Jeffrey Brock ◽  
Christopher Leininger ◽  
Babak Modami ◽  
Kasra Rafi
Keyword(s):  
Closed Surface ◽  
Maximal Dimension ◽  
Curve Complex ◽  
Limit Sets ◽  
Ergodic Measures ◽  
Finite Set ◽  
Geodesic Rays ◽  
Thurston Boundary ◽  
Ending Lamination ◽  
Gromov Boundary

AbstractGiven a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension {3g-3}, for example.We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex.

scholarly journals Infinite Random Matrices and Ergodic Measures

2001 ◽  
Vol 223 (1) ◽  
pp. 87-123 ◽  
Author(s):  
Alexei Borodin ◽  
Grigori Olshanski
Keyword(s):  
Random Matrices ◽  
Ergodic Measures

scholarly journals On the number of ergodic measures for minimal shifts with eventually constant complexity growth

2016 ◽  
Vol 37 (7) ◽  
pp. 2099-2130
Author(s):  
MICHAEL DAMRON ◽  
JON FICKENSCHER
Keyword(s):  
Symbolic Dynamics ◽  
Explicit Description ◽  
Probability Measures ◽  
Bounded Number ◽  
Ergodic Measures ◽  
The One ◽  
Constant Growth ◽  
Block Growth ◽  
Linear Block

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length $n$ relates to the one for words of length $n+1$ for each $n=1,2,3,\ldots \,$.

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