scholarly journals Ruelle operator theorem for non-expansive systems

2009 ◽  
Vol 30 (2) ◽  
pp. 469-487 ◽  
Author(s):  
YUNPING JIANG ◽  
YUAN-LING YE
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Iterated Function Systems ◽  
Ruelle Operator ◽  
Iterated Function

AbstractThe Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

Regularity of Hausdorff measure function for conformal dynamical systems

2016 ◽  
Vol 160 (3) ◽  
pp. 537-563 ◽  
Author(s):  
MARIUSZ URBAŃSKI ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Measure ◽  
Hölder Continuity ◽  
Iterated Function Systems ◽  
Separation Condition ◽  
Holder Continuity ◽  
Measure Function ◽  
Iterated Function ◽  
Strong Separation ◽  
Strong Separation Condition

AbstractWe deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.

scholarly journals Recurrence rates for loosely Markov dynamical systems

2007 ◽  
Vol 82 (1) ◽  
pp. 39-57 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Dynamical Systems ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Iterated Function Systems ◽  
Recurrence Rates ◽  
Exponential Maps ◽  
Iterated Function

AbstractThe concept of loosely Markov dynamical systems is introduced. We show that for these systems the recurrence rates and pointwise dimensions coincide. The systems generated by hyperbolic exponential maps, arbitrary rational functions of the Riemann sphere, and measurable dynamical systems generated by infinite conformal iterated function systems are all checked to be loosely Markov.

Attractors of Infinite Iterated Function Systems Containing Con

2013 ◽  
Vol 59 (2) ◽  
pp. 281-298
Author(s):  
Dan Dumitru
Keyword(s):  
Metric Space ◽  
Existence And Uniqueness ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Infinite Family ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function

Abstract We consider a complete ε-chainable metric space (X, d) and an infinite iterated function system (IIFS) formed by an infinite family of (ε, φ)-functions on X. The aim of this paper is to prove the existence and uniqueness of the attractors of such infinite iterated systems (IIFS) and to give some sufficient conditions for these attractors to be connected. Similar results are obtained in the case when the IIFS is formed by an infinite family of uniformly ε-locally strong Meir-Keeler functions.

Ruelle operator with weakly contractive iterated function systems

2012 ◽  
Vol 33 (4) ◽  
pp. 1265-1290 ◽  
Author(s):  
YUAN-LING YE
Keyword(s):  
Iterated Function Systems ◽  
Potential Functions ◽  
Lipschitz Continuous ◽  
Triple Systems ◽  
Ruelle Operator ◽  
Iterated Function ◽  
Continuous Potential ◽  
Set Up ◽  
Associated Family ◽  
Weakly Contractive

AbstractThe Ruelle operator has been studied extensively both in dynamical systems and iterated function systems (IFSs). Given a weakly contractive IFS $(X, \{w_j\}_{j=1}^m)$ and an associated family of positive continuous potential functions $\{p_j\}_{j=1}^m$, a triple system $(X, \{w_j\}_{j=1}^m, \{p_j\}_{j=1}^m)$is set up. In this paper we study Ruelle operators associated with the triple systems. The paper presents an easily verified condition. Under this condition, the Ruelle operator theorem holds provided that the potential functions are Dini continuous. Under the same condition, the Ruelle operator is quasi-compact, and the iterations sequence of the Ruelle operator converges with a specific geometric rate, if the potential functions are Lipschitz continuous.

Pascal's triangle, dynamical systems and attractors

1992 ◽  
Vol 12 (3) ◽  
pp. 479-486 ◽  
Author(s):  
Fritz V. Haeseler ◽  
Heinz-Otto Peitgen ◽  
Gencho Skordev
Keyword(s):  
Dynamical Systems ◽  
General Framework ◽  
Prime Power ◽  
Systems Approach ◽  
Iterated Function Systems ◽  
Binomial Coefficients ◽  
Self Similarity ◽  
Iterated Function ◽  
Fractal Patterns ◽  
Pascal's Triangle

AbstractThis paper establishes a global dynamical systems approach for the fractal patterns which are obtained when analysing the divisibility of binomial coefficients modulo a prime power. The general framework is within the class of hierarchical iterated function systems. As a consequence we obtain a complete deciphering of the hierarchical self-similarity features.

scholarly journals On topological conjugacy of some chaotic dynamical systems on the Sierpinski gasket

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2317-2331
Author(s):  
Nisa Aslan ◽  
Mustafa Saltan ◽  
Bünyamin Demir
Keyword(s):  
Dynamical Systems ◽  
Sierpinski Gasket ◽  
Iterated Function Systems ◽  
Time Algorithm ◽  
Periodic Points ◽  
Topological Conjugacy ◽  
Escape Time ◽  
Sierpiński Gasket ◽  
Iterated Function ◽  
Self Similar

The dynamical systems on the classical fractals can naturally be obtained with the help of their iterated function systems. In the recent years, different ways have been developed to define dynamical systems on the self similar sets. In this paper, we give composition functions by using expanding and folding mappings which generate the classical Sierpinski Gasket via the escape time algorithm. These functions also indicate dynamical systems on this fractal. We express the dynamical systems by using the code representations of the points. Then, we investigate whether these dynamical systems are topologically conjugate (equivalent) or not. Finally, we show that the dynamical systems are chaotic in the sense of Devaney and then we also compute and compare the periodic points.

A COMPUTATIONAL ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

2008 ◽  
Vol 08 (03) ◽  
pp. 365-381 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
DOAN THAI SON ◽  
STEFAN SIEGMUND
Keyword(s):  
Dynamical Systems ◽  
Euclidean Space ◽  
Ergodic Theorem ◽  
Error Bound ◽  
Metric Space ◽  
Iterated Function Systems ◽  
Random Dynamical Systems ◽  
Compact Metric Space ◽  
Iterated Function ◽  
Time Average

Iterated function systems are examples of random dynamical systems and became popular as generators of fractals like the Sierpinski Gasket and the Barnsley Fern. In this paper we prove an ergodic theorem for iterated function systems which consist of countably many functions and which are contractive on average on an arbitrary compact metric space and we provide a computational version of this ergodic theorem in Euclidean space which allows to numerically approximate the time average together with an explicit error bound. The results are applied to an explicit example.

Chaos in Iterated Function Systems

2020 ◽  
Vol 30 (12) ◽  
pp. 2050177
Author(s):  
Maliheh Mohtashamipour ◽  
Alireza Zamani Bahabadi
Keyword(s):  
Dynamical Systems ◽  
Necessary Conditions ◽  
Iterated Function Systems ◽  
Iterated Function ◽  
Topological Chaos

In the present paper, we study chaos in iterated function systems (IFS), namely dynamical systems with several generators. We introduce weak Li–Yorke chaos, chaos in branch, and weak topological chaos to perceive the role of branches to create chaos in an IFS. Moreover, we define another type of chaos, [Formula: see text]-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function systems.

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