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

Recoding the classical Hénon-Devaney map

Fernando Lenarduzzi

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.
Keywords: Dynamical systems, Hénon-Devaney map, restricted three-body problem, symbolic dynamics, shifts.
Mathematics Subject Classification: Primary: 37D99, 37B10; Secondary: 37B50.
Citation: Fernando Lenarduzzi. Recoding the classical Hénon-Devaney map. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Lenci, On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.  doi: 10.1007/s00220-010-1043-6.  Google Scholar

[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.  Google Scholar

[9]

S. Muñoz, Hyperbolicity and robust transitivity of non-compact invariant sets for the plane, preprint. Google Scholar

[10]

E. Pujals and F. Lenarduzzi, Generalized hénon-devaney maps, work in progress. Google Scholar

[11]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Lenci, On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.  doi: 10.1007/s00220-010-1043-6.  Google Scholar

[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.  Google Scholar

[9]

S. Muñoz, Hyperbolicity and robust transitivity of non-compact invariant sets for the plane, preprint. Google Scholar

[10]

E. Pujals and F. Lenarduzzi, Generalized hénon-devaney maps, work in progress. Google Scholar

[11]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

Figure 1.  The map $ f $ and the discontinuity
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Figure 2.  The dynamics of the part above $ R_1 $
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Table 
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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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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