Cardinality of survival set for the chaotic Tent map with holes.

The aim of this paper is to give an overview of the dynamics of one dimensionaldiscrete dynamical systems: Tent map family T3, Doubling map E2and shift map σ are investigated. Let I-intervals(Holes) lie in the interval [0,1) and let E2be a Doubling map. The survivor set Ω(I) :={x∈[0,1) :Enx /∈ I, n≥0}. Depending on location and size of the intervals we will characterize the survivor set Ω(I) infinite or finite. Also we will show conjugacy of some maps that used in this paper. By using conjugacy of functions we will show that the Survivor set is infinite or finite in another composition of maps. The Cantor sets Λ that occur as non-survivor sets for c >2 from Tent map family Tc.[1] Keywords: dynamical systems, symbolic dynamics, interval maps, survivor sets, chaos, open systems, irregular sets.

The -transformation with a hole at 0

For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by $$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$ In this paper we characterize the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ of all parameters $t\in [0,1)$ for which the set-valued function $t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$ is not locally constant. We show that $E_{\unicode[STIX]{x1D6FD}}$ is a Lebesgue null set of full Hausdorff dimension for all $\unicode[STIX]{x1D6FD}\in (1,2)$. We prove that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$ the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\unicode[STIX]{x1D6FD}\in (1,2)$ for which $E_{\unicode[STIX]{x1D6FD}}$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_{2}$, the bifurcation set of the doubling map. Finally, we give for each $\unicode[STIX]{x1D6FD}\in (1,2)$ a lower and an upper bound for the value $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$ such that the Hausdorff dimension of $K_{\unicode[STIX]{x1D6FD}}(t)$ is positive if and only if $t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$. We show that $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$ for all $\unicode[STIX]{x1D6FD}\in (1,2)$.

On–off intermittency and chaotic walks

We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on–off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.

MULTIFRACTAL ANALYSIS OF THE BIRKHOFF SUMS OF SAINT-PETERSBURG POTENTIAL

Let [Formula: see text] be the doubling map in the unit interval and [Formula: see text] be the Saint-Petersburg potential, defined by [Formula: see text] if [Formula: see text] for all [Formula: see text]. We consider asymptotic properties of the Birkhoff sum [Formula: see text]. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that [Formula: see text] converges to [Formula: see text] in probability. We determine the Hausdorff dimension of the level set [Formula: see text], as well as that of the set [Formula: see text], when [Formula: see text], [Formula: see text] or [Formula: see text] for [Formula: see text]. The fast increasing Birkhoff sum of the potential function [Formula: see text] is also studied.

Sensitive Dependence on Initial Conditions for an Example of Markov Map: Skewed Doubling Map

Markov map is one example of interval maps where it is a piecewise ex-panding map and obeys the Markov property. One well-known example of Markov map is the doubling map, a map which has two subintervals with equal partitions. In this paper, we are interested to investigate another type of Markov map, the so-called skewed doubling map. This map is a more generalized map than the doubling map. Thus, the aims of this paper are to nd the xed points as well as the periodic points for the skewed doubling map and to investigate the sensitive dependence on initial conditions of this map. The method considered here is the cobweb diagram. Numerical results suggest that there exist dense of periodic orbits for this map. The sensitivity of this map to initial conditions is also veried where small differences in initial conditions give dierent behaviour of the orbits in the map.

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

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.

Interactions of the Julia Set with Critical and (Un)Stable Sets in an Angle-Doubling Map on ℂ\{0}

We study a nonanalytic perturbation of the complex quadratic family z ↦ z2 + c in the form of a two-dimensional noninvertible map that has been introduced by Bamón et al. [2006]. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimages. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. For parameters away from the complex quadratic family we define a generalized notion of the Julia set as the basin boundary of infinity. We are interested in how the Julia set changes when saddle points along with their stable and unstable sets appear as the perturbation is switched on. Advanced numerical techniques enable us to study the interactions of the Julia set with the critical set and the (un)stable sets of saddle points. We find the appearance and disappearance of chaotic attractors and dramatic changes in the topology of the Julia set; these bifurcations lead to three complicated types of Julia sets that are given by the closure of stable sets of saddle points of the map, namely, a Cantor bouquet and what we call a Cantor tangle and a Cantor cheese. We are able to illustrate how bifurcations of the nonanalytic map connect to those of the complex quadratic family by computing two-parameter bifurcation diagrams that reveal a self-similar bifurcation structure near the period-doubling route to chaos in the complex quadratic family.