scholarly journals Lp-GENERIC COCYCLES HAVE ONE-POINT LYAPUNOV SPECTRUM

2003 ◽  
Vol 03 (01) ◽  
pp. 73-81 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
JAIRO BOCHI
Keyword(s):  
Lyapunov Exponents ◽  
Lyapunov Spectrum ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Function

We show that the sum of the first k Lyapunov exponents of linear cocycles is an upper semicontinuous function in the Lp topologies, for any 1 ≤ p ≤ ∞ and k. This fact, together with a result from Arnold and Cong, implies that the Lyapunov exponents of the Lp-generic cocycle, p < ∞, are all equal.

scholarly journals PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 210-229
Author(s):  
O. Maslyuchenko ◽  
A. Kushnir
Keyword(s):  
Continuous Function ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Normal Space ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Pseudocompact Space ◽  
Upper Semicontinuous Function ◽  
Perfectly Normal Space ◽  
Perfectly Normal

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .

CARDINALITY OF INVERSE LIMITS WITH UPPER SEMICONTINUOUS BONDING FUNCTIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 167-174 ◽  
Author(s):  
MATEJ ROŠKARIČ ◽  
NIKO TRATNIK
Keyword(s):  
Recent Result ◽  
Inverse Limit ◽  
Inverse Limits ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Function

AbstractWe explore the cardinality of generalised inverse limits. Among other things, we show that, for any $n\in \{&#x2135;_{0},c,1,2,3,\dots \}$, there is an upper semicontinuous function with the inverse limit having exactly $n$ points. We also prove that if $f$ is an upper semicontinuous function whose graph is a continuum, then the cardinality of the corresponding inverse limit is either 1, $&#x2135;_{0}$ or $c$. This generalises the recent result of I. Banič and J. Kennedy, which claims that the same is true in the case where the graph is an arc.

scholarly journals Extremal $\omega$-plurisubharmonic functions as envelopes of disc functionals: generalization and applications to the local theory

2012 ◽  
Vol 111 (2) ◽  
pp. 296
Author(s):  
Benedikt Steinar Magnússon
Keyword(s):  
Complex Manifold ◽  
Plurisubharmonic Function ◽  
Local Theory ◽  
Semicontinuous Function ◽  
Plurisubharmonic Functions ◽  
The Real ◽  
Upper Semicontinuous ◽  
The Difference ◽  
Upper Semicontinuous Function

We generalize the Poletsky disc envelope formula for the function $\sup \{u\in \mathcal{PSH}(X,\omega); u\leq \phi\}$ on any complex manifold $X$ to the case where the real $(1,1)$-current $\omega=\omega_1-\omega_2$ is the difference of two positive closed $(1,1)$-currents and $\varphi$ is the difference of an $\omega_1$-upper semicontinuous function and a plurisubharmonic function.

scholarly journals Riesz's Functions in Weighted Hardy and Bergman Spaces

1996 ◽  
Vol 48 (5) ◽  
pp. 930-945 ◽  
Author(s):  
Takahiko Nakazi ◽  
Masahiro Yamada
Keyword(s):  
Hardy Spaces ◽  
Borel Measure ◽  
Bergman Spaces ◽  
Semicontinuous Function ◽  
Positive Borel Measure ◽  
Upper Semicontinuous ◽  
Hardy And Bergman Spaces ◽  
Upper Semicontinuous Function ◽  
Image Position ◽  
Bergman And Hardy Spaces

AbstractLet μ be a finite positive Borel measure on the closed unit disc . For each a in , put where ƒ ranges over all analytic polynomials with f(a) = 1. This upper semicontinuous function S(a) is called a Riesz's function and studied in detail. Moreover several applications are given to weighted Bergman and Hardy spaces.

scholarly journals Super-extremal processes and the argmax process

1994 ◽  
Vol 31 (04) ◽  
pp. 958-978 ◽  
Author(s):  
Sidney I. Resnick ◽  
Rishin Roy
Keyword(s):  
Metric Space ◽  
Semicontinuous Function ◽  
Closed Set ◽  
Continuous Choice ◽  
Extremal Processes ◽  
Upper Semicontinuous Function ◽  
Classical Vector ◽  
Vector Valued ◽  
Path Properties ◽  
Separable Metric Space

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processes Y = {Yt, t &gt; 0}. At any t &gt; 0, Y t is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t &gt; 0, Y t is associated. For a compact metric E, we consider the argmax process M = {Mt, t &gt; 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1

2012 ◽  
Vol 12 (04) ◽  
pp. 1250002 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
NGUYEN THI THE
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Exponents ◽  
Algebraic Equation ◽  
Differential Algebraic Equations ◽  
Lyapunov Spectrum ◽  
Stochastic Flow ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Two Parameter

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

scholarly journals Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

2019 ◽  
Vol 26 (3) ◽  
pp. 195-209 ◽  
Author(s):  
Frank Kwasniok
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Finite Time ◽  
Large Deviation ◽  
Atmospheric Model ◽  
Lyapunov Spectrum ◽  
Finite Time Lyapunov Exponents ◽  
Fluctuation Field ◽  
Deviation Rate ◽  
Rate Functions

Abstract. The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

scholarly journals Super-extremal processes and the argmax process

1994 ◽  
Vol 31 (4) ◽  
pp. 958-978 ◽  
Author(s):  
Sidney I. Resnick ◽  
Rishin Roy
Keyword(s):  
Metric Space ◽  
Semicontinuous Function ◽  
Closed Set ◽  
Continuous Choice ◽  
Extremal Processes ◽  
Upper Semicontinuous Function ◽  
Classical Vector ◽  
Vector Valued ◽  
Path Properties ◽  
Separable Metric Space

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processesY= {Yt, t > 0}. At any t > 0, Yt is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t > 0, Yt is associated.For a compact metric E, we consider the argmax process M = {Mt, t > 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.

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