scholarly journals The local Hölder exponent for the dimension of invariant subsets of the circle

2016 ◽  
Vol 37 (6) ◽  
pp. 1825-1840 ◽  
Author(s):  
CARLO CARMINATI ◽  
GIULIO TIOZZO
Keyword(s):  
Topological Entropy ◽  
Dimension Function ◽  
Bifurcation Parameter ◽  
Similar Statement ◽  
Hölder Exponent ◽  
Expanding Maps ◽  
Forward Orbit ◽  
Doubling Map

We consider for each $t$ the set $K(t)$ of points of the circle whose forward orbit for the doubling map does not intersect $(0,t)$, and look at the dimension function $\unicode[STIX]{x1D702}(t):=\text{H.dim}\,K(t)$. We prove that at every bifurcation parameter $t$, the local Hölder exponent of the dimension function equals the value of the function $\unicode[STIX]{x1D702}(t)$ itself. A similar statement holds for general expanding maps of the circle: namely, we consider the topological entropy of the map restricted to the survival set, and obtain bounds on its local Hölder exponent in terms of the value of the function.

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

scholarly journals On Hausdorff dimension of invariant sets for expanding maps of a circle

1986 ◽  
Vol 6 (2) ◽  
pp. 295-309 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Cantor Set ◽  
Invariant Sets ◽  
Expanding Maps ◽  
Cantor Function ◽  
The Family ◽  
Expanding Mapping

AbstractGiven an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).

Dynamical upper bounds for Hausdorff dimension of invariant sets

1997 ◽  
Vol 17 (3) ◽  
pp. 739-756 ◽  
Author(s):  
YINGJIE ZHANG
Keyword(s):  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Upper Bounds ◽  
Invariant Sets ◽  
Topological Pressure ◽  
Expanding Maps ◽  
Unstable Manifolds ◽  
Hyperbolic Sets

We study the Hausdorff dimension of invariant sets for expanding maps and that of hyperbolic sets on unstable manifolds. Upper bounds for the Hausdorff dimension are given in terms of topological pressure, or topological entropy and Lyapunov exponents.

scholarly journals A note on periodic points of expanding maps of the interval

1986 ◽  
Vol 33 (3) ◽  
pp. 435-447 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Points ◽  
Topological Entropy ◽  
Periodic Points ◽  
Expanding Maps ◽  
Continuous Maps

We sharpen a result of Byers on the existence of periodic points for some continuous expanding maps of the interval and generalize it to some classes of continuous maps of the interval which are not necessarily expanding. We then use these results to construct one-parameter families of continuous maps of the interval which have a bifurcation form fixed points directly to period 3 points together with a series of reverse bifurcations from period 3 points back to fixed points. Consequently, our results also provide examples of one-parameter families of continuous maps of the interval whose topological entropy jumps form zero to some positive number and then changes back to zero as the parameter varies.

scholarly journals Entropy continuity for interval maps with holes

2017 ◽  
Vol 38 (6) ◽  
pp. 2036-2061 ◽  
Author(s):  
OSCAR F. BANDTLOW ◽  
HANS HENRIK RUGH
Keyword(s):  
Topological Entropy ◽  
Fixed Size ◽  
Hölder Exponent ◽  
Interval Maps ◽  
Uniform Rate ◽  
Piecewise Monotonic

We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole at a uniform rate. We show that under suitable conditions the topological entropy varies locally Hölder continuously with the local Hölder exponent depending itself on the value of the topological entropy.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Transmission rate conditions for distributed filtering in sensor networks against eavesdropper

2021 ◽  
pp. 014233122110056
Author(s):  
Bingya Zhao ◽  
Ya Zhang
Keyword(s):  
Sensor Networks ◽  
Topological Entropy ◽  
Kalman Filtering ◽  
Transmission Rate ◽  
Necessary Condition ◽  
Estimation Errors ◽  
Positive Role ◽  
Filtering Algorithm ◽  
Wiretap Channels ◽  
Kalman Filtering Algorithm

This paper studies the distributed secure estimation problem of sensor networks (SNs) in the presence of eavesdroppers. In an SN, sensors communicate with each other through digital communication channels, and the eavesdropper overhears the messages transmitted by the sensors over fading wiretap channels. The increasing transmission rate plays a positive role in the detectability of the network while playing a negative role in the secrecy. Two types of SNs under two cooperative filtering algorithms are considered. For networks with collectively observable nodes and the Kalman filtering algorithm, by studying the topological entropy of sensing measurements, a sufficient condition of distributed detectability and secrecy, under which there exists a code–decode strategy such that the sensors’ estimation errors are bounded while the eavesdropper’s error grows unbounded, is given. For collectively observable SNs under the consensus Kalman filtering algorithm, by studying the topological entropy of the sensors’ covariance matrices, a necessary condition of distributed detectability and secrecy is provided. A simulation example is given to illustrate the results.

scholarly journals Entropy on noncompact sets

2019 ◽  
Vol 7 (1) ◽  
pp. 29-37
Author(s):  
Jose S. Cánovas
Keyword(s):  
Topological Entropy ◽  
Topological Spaces ◽  
Continuous Maps

AbstractIn this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.

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