Figure 1.
The map $ f $ and the discontinuity
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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 |
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 and Continuous Dynamical Systems,
2020, 40
(7)
: 4073-4092.
doi: 10.3934/dcds.2020172
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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. |
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. |
| [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. |
Figure 2.
The dynamics of the part above $ R_1 $
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