# Topological dynamics of the doubling map with asymmetrical holes

@article{Barrera2015TopologicalDO, title={Topological dynamics of the doubling map with asymmetrical holes}, author={Rafael Alcaraz Barrera}, journal={arXiv: Dynamical Systems}, year={2015} }

We study the dynamics of the attractor of the doubling map with an asymmetrical hole by associating to each hole an element of the lexicographic world. A description of the topological entropy function is given. We show that the set of parameters $(a,b)$ such that the dynamics of the mentioned attractor corresponds to a subshift of finite type is open and dense. Using the connections between this family of open dynamical systems, intermediate $\beta$-expansions and Lorenz maps we study the…

## 2 Citations

Intrinsic Ergodicity of Open dynamical systems for the doubling map

- Mathematics
- 2015

We give sufficient conditions for intervals $(a,b)$ such that the associated open dynamical system for the doubling map is intrinsically ergodic. We also show that the set of parameters $(a,b) \in…

The k-Transformation on an Interval with a Hole

- Mathematics
- 2017

Let $T_{k}$ be the expanding map of $[0,1)$ defined by $T_{k}(x) = k x\ \text{mod 1}$, where $k\geq 2$ is an integer. Given $0\leq a<b\leq 1$, let $\mathcal{W}_{k}(a,b)=\{x\in [0,1)\ \vert \…

## References

SHOWING 1-10 OF 36 REFERENCES

Topological and symbolic dynamics of the doubling map with a hole

- Mathematics
- 2014

This work motivates the study of open dynamical systems corresponding to the doubling map. In particular, the dynamical properties of the attractor of the doubling map when a symmetric, centred open…

Topological and symbolic dynamics for hyperbolic systems with holes

- MathematicsErgodic Theory and Dynamical Systems
- 2010

Abstract We consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Ω* of orbits which never falls into a fixed hole. We study various aspects of the symbolic…

Essential Dynamics for Lorenz maps on the real line and the Lexicographical World ? ? Partially supp

- Mathematics
- 2006

Abstract In this paper we describe some topological and geometric properties of the set of sequences LW = { ( a , b ) ∈ Σ 0 × Σ 1 ; a ⩽ σ n ( a ) ⩽ b , a ⩽ σ n ( b ) ⩽ b , ∀ n ∈ N } , which…

Open circle maps: Small hole asymptotics

- Mathematics, Physics
- 2011

We consider escape from chaotic maps through a subset of phase space, the hole. Escape rates are known to be locally constant functions of the hole position and size. In spite of this, for the…

Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps

- Mathematics
- 1993

Abstract We study the dynamics and kneading theory of topologically expansive piecewise increasing maps of the interval with a single discontinuity (Lorenz maps). We characterise those maps which are…

Zeros of the kneading invariant and topological entropy for Lorenz maps

- Mathematics
- 1996

If is a unimodal map, then its topological entropy is related to the smallest positive zero s of a certain power series (the kneading invariant of f) by . Moreover, it is implicit in the results of…

The classification of topologically expansive lorenz maps

- Mathematics
- 1990

We show that topologically expansive Lorenz maps can be described up to topological conjugacy by their kneading invariants. We also give a simple condition on pairs of symbol sequences which is…

Topological conjugation of Lorenz maps by β-transformations

- Mathematics
- 1990

Necessary and sufficient conditions for a Lorenz map to be topologically conjugate to a piecewise linear map with constant slope (a β-transformation) are given, first in terms of kneading invariants…

Topological and Ergodic properties of symmetric subshifts

- Mathematics
- 2013

The family of symmetric one sided subshifts in two symbols given by a sequence $a$ is studied. We analyse some of their topological properties such as transitivity, the specification property and…

Invariant subsets of expanding mappings of the circle

- Mathematics
- 1987

The continuity of Hausdorff dimension of closed invariant subsets K of a C 2 -expanding mapping g of the circle is investigated. If g / K satisfies the specification property then the equilibrium…