BIRKHOFF SPECTRA FOR ONE-DIMENSIONAL MAPS WITH SOME HYPERBOLICITY

2010 ◽  
Vol 10 (01) ◽  
pp. 53-75 ◽  
Author(s):  
YONG MOO CHUNG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Probability Measures ◽  
One Dimensional ◽  
System A ◽  
Topologically Mixing ◽  
One Dimensional Maps ◽  
Dimension One

We study the multifractal analysis for smooth dynamical systems in dimension one. It is given a characterization of the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing C2 map modeled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.

scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

BORDER COLLISION BIFURCATIONS IN 1D PWL MAP WITH ONE DISCONTINUITY AND NEGATIVE JUMP: USE OF THE FIRST RETURN MAP

2010 ◽  
Vol 20 (11) ◽  
pp. 3529-3547 ◽  
Author(s):  
LAURA GARDINI ◽  
FABIO TRAMONTANA
Keyword(s):  
Parameter Space ◽  
Complete Characterization ◽  
One Dimensional ◽  
Return Map ◽  
Border Collision Bifurcations ◽  
One Dimensional Maps ◽  
Negative Jump

The aim of this work is to study discontinuous one-dimensional maps in the case of slopes and offsets having opposite signs. Such models represent the dynamics of applied systems in several disciplines. We analyze in particular attracting cycles, their border collision bifurcations and the properties of the periodicity regions in the parameter space. The peculiarity of this family is that we can make use of the technical instrument of the first return map. With this, we can rigorously prove properties which were known numerically, as well as prove new ones, giving a complete characterization of the overlapping periodicity regions.

scholarly journals On the notion of the dimension with respect to a dynamical system

1984 ◽  
Vol 4 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Invariant Sets ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Common Features ◽  
Dynamical Properties ◽  
The One ◽  
Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

scholarly journals A classification of explosions in dimension one

2009 ◽  
Vol 29 (2) ◽  
pp. 715-731 ◽  
Author(s):  
E. SANDER ◽  
J. A. YORKE
Keyword(s):  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Necessary And Sufficient Conditions ◽  
Homoclinic Tangency ◽  
Discontinuous Change ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Dimension One ◽  
Necessary And Sufficient

AbstractA discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, anexplosionis a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

MULTIFRACTAL ANALYSIS OF ONE-DIMENSIONAL BIASED WALKS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850030 ◽  
Author(s):  
YUFEI CHEN ◽  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YU SUN ◽  
WEIYI SU
Keyword(s):  
Random Walk ◽  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Infinite Sequence ◽  
Real Numbers ◽  
One Dimensional ◽  
Dimension Formula ◽  
Sum Of Digits ◽  
Packing Dimensions ◽  
Dyadic System

For an infinite sequence [Formula: see text] of [Formula: see text] and [Formula: see text] with probability [Formula: see text] and [Formula: see text], we mainly study the multifractal analysis of one-dimensional biased walks. Let [Formula: see text] and [Formula: see text]. The Hausdorff and packing dimensions of the sets [Formula: see text] are [Formula: see text], which is the development of the theorem of Besicovitch [On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1934) 321–330] on random walk, saying that: For any [Formula: see text], the set [Formula: see text] has Hausdorff dimension [Formula: see text].

scholarly journals The Fourier–Stieltjes algebra of a C*-dynamical system

2016 ◽  
Vol 27 (06) ◽  
pp. 1650050 ◽  
Author(s):  
Erik Bédos ◽  
Roberto Conti
Keyword(s):  
Dynamical System ◽  
Banach Algebra ◽  
Type Theorem ◽  
Positive Definiteness ◽  
Crossed Product ◽  
System Building ◽  
System A ◽  
Commutative Subalgebras ◽  
Definition Of

In analogy with the Fourier–Stieltjes algebra of a group, we associate to a unital discrete twisted [Formula: see text]-dynamical system a Banach algebra whose elements are coefficients of equivariant representations of the system. Building upon our previous work, we show that this Fourier–Stieltjes algebra embeds continuously in the Banach algebra of completely bounded multipliers of the (reduced or full) [Formula: see text]-crossed product of the system. We introduce a notion of positive definiteness and prove a Gelfand–Raikov type theorem allowing us to describe the Fourier–Stieltjes algebra of a system in a more intrinsic way. We also propose a definition of amenability for [Formula: see text]-dynamical systems and show that it implies regularity. After a study of some natural commutative subalgebras, we end with a characterization of the Fourier–Stieltjes algebra involving [Formula: see text]-correspondences over the (reduced or full) [Formula: see text]-crossed product.

scholarly journals Characterization of Low Dimensional RCD*(K, N) Spaces

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Hausdorff Dimension ◽  
Isoperimetric Inequality ◽  
Type Inequality ◽  
Metric Measure Spaces ◽  
One Dimensional ◽  
Curvature Bounds ◽  
Measure Spaces ◽  
Low Dimensional ◽  
Regular Sets

AbstractIn this paper,we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD*(K, N) spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdorff dimension N and the class of RCD*(K, N) spaces coincide for N < 2 (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse to the Lévy-Gromov’s isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

Entropy in dimension one

2014 ◽  
Author(s):  
William P. Thurston
Keyword(s):  
Topological Entropy ◽  
Finite Interval ◽  
Algebraic Integer ◽  
Small Letter ◽  
One Dimensional ◽  
Original Manuscript ◽  
Postcritically Finite ◽  
Perron Number ◽  
One Dimensional Maps ◽  
Dimension One

This chapter studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eʰ, presenting some constructions for maps with given entropy and characterizing what values of entropy can occur for postcritically finite maps. In particular, the chapter proves: h is the topological entropy of a postcritically finite interval map if and only if h = log λ‎, where λ‎ ≥ 1 is a weak Perron number, i.e., it is an algebraic integer, and λ‎ ≥ ∣λ‎superscript Greek Small Letter Sigma∣ for every Galois conjugate λ‎superscript Greek Small Letter Sigma ∈ C. Unfortunately, the author of this chapter has died before completing this work, hence this chapter contains both the original manuscript as well as a number of notes which clarify many of the points mentioned therein.

scholarly journals Type-II intermittency in a class of two coupled one-dimensional maps

2000 ◽  
Vol 5 (3) ◽  
pp. 233-245 ◽  
Author(s):  
J. Laugesen ◽  
E. Mosekilde ◽  
T. Bountis ◽  
S. P. Kuznetsov
Keyword(s):  
Period Doubling ◽  
Coupled System ◽  
Two Dimensions ◽  
Type Ii ◽  
One Dimensional ◽  
System A ◽  
Subcritical Hopf Bifurcation ◽  
Theoretical Predictions ◽  
The Individual ◽  
One Dimensional Maps

The paper shows how intermittency behavior of type-II can arise from the coupling of two one-dimensional maps, each exhibiting type-III intermittency. This change in dynamics occurs through the replacement of a subcritical period-doubling bifurcation in the individual map by a subcritical Hopf bifurcation in the coupled system. A variety of different parameter combinations are considered, and the statistics for the distribution of laminar phases is worked out. The results comply well with theoretical predictions. Provided that the reinjection process is reasonably uniform in two dimensions, the transition to type-II intermittency leads directly to higher order chaos. Hence, this transition represents a universal route to hyperchaos.

Sign in / Sign up

Export Citation Format

Share Document