# American Institute of Mathematical Sciences

July  2020, 40(7): 4073-4092. doi: 10.3934/dcds.2020172

## Recoding the classical Hénon-Devaney map

 Universidade Federal de São Carlos, São Carlos, SP - 13565905, Brazil

Received  May 2018 Revised  December 2019 Published  April 2020

Fund Project: The author is supported by CAPES

In this work we are going to consider the classical Hénon-Devaney map given by
 $\begin{eqnarray*} f: \mathbb{R}^2\setminus \{y = 0\} &\rightarrow& \mathbb{R}^2 \\ (x,y) &\mapsto& \left(x+\dfrac{1}{y}, y-\dfrac{1}{y}-x\right) \end{eqnarray*}$
We are going to construct conjugacy to a subshift of finite type, providing a global understanding of the map's behavior.We extend the coding to a more general class of maps that can be seen as a map in a square with a fixed discontinuity.
Citation: Fernando Lenarduzzi. Recoding the classical Hénon-Devaney map. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4073-4092. doi: 10.3934/dcds.2020172
##### References:

show all references

##### References:
The map $f$ and the discontinuity
The dynamics of the part above $R_1$
 Initial Point $p$ $f(p)$ $f^2(p)$ coordinates $(3\oplus -2, 1\oplus -4)$ $(2\oplus -2, 2\oplus -4)$ $(1\oplus -2, 3\oplus -4)$ coding $(2 \ ,\ 1)$ $(22 \ , \ 12)$ $(221 \ , \ 122)$ coordinates $(-1\oplus 3\oplus -1, -1)$ $( 3\oplus -1, -2)$ $( 2\oplus -1, 1\oplus -2)$ coding $(-1 \ , \ -1)$ $(-12 \ , \ -1-2)$ $(\operatorname{-122} \ , \ \operatorname{-1-21})$ coordinates $(-1\oplus 1\oplus -1, -1)$ $( 1\oplus -1, -2)$ $( -1, 1\oplus -2)$ coding $(-1 \ , \ -1)$ $(\operatorname{-11} \ , \ \operatorname{-1-2})$ $(\operatorname{-11-1} \ , \ \operatorname{-1-21})$ coordinates $(3\oplus -2, 3\oplus -4)$ $(2\oplus -2, 4\oplus -4)$ $(1\oplus -2, 5\oplus -4)$ coding $(2 \ ,\ 2)$ $(22 \ , \ 22)$ $(221 \ , \ 222)$ coordinates $(3\oplus -2, -1\oplus 4)$ $(2\oplus -2, 1\oplus-1\oplus -4)$ $(1\oplus -2, 2\oplus-1\oplus -4)$ coding $(2 \ ,\ \operatorname{-1})$ $(22 \ , \ \operatorname{-11})$ $(221 \ , \ \operatorname{-112})$
 Initial Point $p$ $f(p)$ $f^2(p)$ coordinates $(3\oplus -2, 1\oplus -4)$ $(2\oplus -2, 2\oplus -4)$ $(1\oplus -2, 3\oplus -4)$ coding $(2 \ ,\ 1)$ $(22 \ , \ 12)$ $(221 \ , \ 122)$ coordinates $(-1\oplus 3\oplus -1, -1)$ $( 3\oplus -1, -2)$ $( 2\oplus -1, 1\oplus -2)$ coding $(-1 \ , \ -1)$ $(-12 \ , \ -1-2)$ $(\operatorname{-122} \ , \ \operatorname{-1-21})$ coordinates $(-1\oplus 1\oplus -1, -1)$ $( 1\oplus -1, -2)$ $( -1, 1\oplus -2)$ coding $(-1 \ , \ -1)$ $(\operatorname{-11} \ , \ \operatorname{-1-2})$ $(\operatorname{-11-1} \ , \ \operatorname{-1-21})$ coordinates $(3\oplus -2, 3\oplus -4)$ $(2\oplus -2, 4\oplus -4)$ $(1\oplus -2, 5\oplus -4)$ coding $(2 \ ,\ 2)$ $(22 \ , \ 22)$ $(221 \ , \ 222)$ coordinates $(3\oplus -2, -1\oplus 4)$ $(2\oplus -2, 1\oplus-1\oplus -4)$ $(1\oplus -2, 2\oplus-1\oplus -4)$ coding $(2 \ ,\ \operatorname{-1})$ $(22 \ , \ \operatorname{-11})$ $(221 \ , \ \operatorname{-112})$
 [1] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [2] Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349 [3] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [4] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [5] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [6] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [7] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [8] Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116 [9] Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426 [10] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [11] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [12] Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449 [13] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381 [14] Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018 [15] Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012 [16] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [17] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [18] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [19] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [20] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

2019 Impact Factor: 1.338