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 and Continuous Dynamical Systems, 2020, 40 (7) : 4073-4092. doi: 10.3934/dcds.2020172
References:
 [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050. [2] R. L. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel J. Math., 16 (1973), 263-278.  doi: 10.1007/BF02756706. [3] P. Cirilo, Y. Lima and E. Pujals, Ergodic properties of skew products in infinite measure, Israel J. Math., 214 (2016), 43-66.  doi: 10.1007/s11856-016-1344-3. [4] R. L. Devaney, The baker transformation and a mapping associated to the restricted three-body problem, Commun. Math. Phys., 80 (1981), 465-476.  doi: 10.1007/BF01941657. [5] M. Hénon, Generating Families in the Restricted Three-Body Problem, Lecture Notes in Physics, New Series m: Monographs, 52, Springer-Verlag, Berlin, 1997. doi: 10.1007/3-540-69650-4. [6] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042. [7] M. Lenci, On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.  doi: 10.1007/s00220-010-1043-6. [8] S. Muñoz, Robust transitivity of maps of the real line, Discrete Contin. Dyn. Syst., 35 (2015), 1163-1177.  doi: 10.3934/dcds.2015.35.1163. [9] S. Muñoz, Hyperbolicity and robust transitivity of non-compact invariant sets for the plane, preprint. [10] E. Pujals and F. Lenarduzzi, Generalized hénon-devaney maps, work in progress. [11] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

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References:
 [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050. [2] R. L. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel J. Math., 16 (1973), 263-278.  doi: 10.1007/BF02756706. [3] P. Cirilo, Y. Lima and E. Pujals, Ergodic properties of skew products in infinite measure, Israel J. Math., 214 (2016), 43-66.  doi: 10.1007/s11856-016-1344-3. [4] R. L. Devaney, The baker transformation and a mapping associated to the restricted three-body problem, Commun. Math. Phys., 80 (1981), 465-476.  doi: 10.1007/BF01941657. [5] M. Hénon, Generating Families in the Restricted Three-Body Problem, Lecture Notes in Physics, New Series m: Monographs, 52, Springer-Verlag, Berlin, 1997. doi: 10.1007/3-540-69650-4. [6] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042. [7] M. Lenci, On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.  doi: 10.1007/s00220-010-1043-6. [8] S. Muñoz, Robust transitivity of maps of the real line, Discrete Contin. Dyn. Syst., 35 (2015), 1163-1177.  doi: 10.3934/dcds.2015.35.1163. [9] S. Muñoz, Hyperbolicity and robust transitivity of non-compact invariant sets for the plane, preprint. [10] E. Pujals and F. Lenarduzzi, Generalized hénon-devaney maps, work in progress. [11] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
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 $(2 \ ,\ \operatorname{-1})$ $(22 \ , \ \operatorname{-11})$ $(221 \ , \ \operatorname{-112})$
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