Figure 1.
Graph of a generic idempotent continuous function in $ \mathbb R $
-
Previous Article
Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions
- DCDS Home
- This Issue
-
Next Article
Well-posedness of some non-linear stable driven SDEs
February
2021, 41(2): 899-919.
doi: 10.3934/dcds.2020303
A generalization of the Babbage functional equation
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom |
A recent refinement of Kerékjártó's Theorem has shown that in $ \mathbb R $ and $ \mathbb R^2 $ all $ \mathcal C^l $–solutions of the functional equation $ f^n = \text{Id} $ are $ \mathcal C^l $–linearizable, where $ l\in \{0,1,\dots \infty\} $. When $ l\geq 1 $, in the real line we prove that the same result holds for solutions of $ f^n = f $, while we can only get a local version of it in the plane. Through examples, we show that these results are no longer true when $ l = 0 $ or when considering the functional equation $ f^n = f^k $ with $ n>k\geq 2 $.
Keywords:
Babbage functional equation,
functional equation,
Kerékjártó theorem,
periodic map,
idempotent map,
linearization,
topological conjugacy,
Hardy–Weinberg equilibrium.
Mathematics Subject Classification:
Primary: 37C15, 39B12; Secondary: 30D05, 37C05, 37E05, 37E30, 39B22.
Citation:
Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete and Continuous Dynamical Systems,
2021, 41
(2)
: 899-919.
doi: 10.3934/dcds.2020303
![]()
References:
| [1] |
C. Babbage,
An essay towards the calculus of functions, Philos. Trans. Royal Soc., 105 (1815), 389-423.
|
| [2] |
N. Bacaër, A Short History of Mathematical Population Dynamics, Springer-Verlag London Ltd., London, 2011.
doi: 10.1007/978-0-85729-115-8. |
| [3] |
K. Baron and W. Jarczyk,
Recent results on functional equations in a single variable, perspectives and open problems, Aequ. Math., 61 (2001), 1-48.
doi: 10.1007/s000100050159. |
| [4] |
R. H. Bing,
Inequivalent families of periodic homeomorphisms of $E^3$, Ann. of Math., 80 (1964), 78-93.
doi: 10.2307/1970492. |
| [5] |
A. Cima, A. Gasull, F. Mañosas and R. Ortega,
Linearization of planar involutions in $\mathcal C^1$, Ann. Mat. Pura Appl., 194 (2015), 1349-1357.
doi: 10.1007/s10231-014-0423-5. |
| [6] |
A. Cima, A. Gasull, F. Mañosas and R. Ortega,
Smooth linearisation of planar periodic maps, Math. Proc. Camb. Philos. Soc., 167 (2019), 295-320.
doi: 10.1017/S0305004118000336. |
| [7] |
A. Constantin and B. Kolev,
The theorem of Kerérekjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math., 40 (1994), 193-204.
|
| [8] |
G. M. Ewing and W. R. Utz,
Continuous solutions of the functional equation $f^n(x) = f(x)$, Canadian J. Math., 5 (1953), 101-103.
doi: 10.4153/CJM-1953-012-8. |
| [9] |
A. Haefliger,
Plongements différentiables de variétés dans variétés, Comment. Math. Helv., 36 (1962), 47-82.
|
| [10] |
R. Haynes, S. Kwasik, J. Mast and R. Schultz,
Periodic maps on $R^7$ without fixed points, Math. Proc. Camb. Philos. Soc., 132 (2002), 131-136.
doi: 10.1017/S0305004101005345. |
| [11] |
M. Hirsch, Differential Topology, Springer-Verlag, 1976. |
| [12] |
M. Holz, K. Steffens and E. Weitz, Introduction to Cardinal Arithmetic, Birkhäuser Verlag, 2010.
doi: 10.1007/978-3-0346-0330-0. |
| [13] |
G. Ishikawa and T. Nishimura,
Smooth retracts of Euclidean space, Kodai Math. J., 18 (1995), 260-265.
|
| [14] |
W. Jarczyk,
Babbage equation on the circle, Publ. Math., 63 (2003), 389-400.
|
| [15] |
N. McShane,
On the periodicity of homeomorphisms of the real line, Amer. Math. Monthly, 68 (1961), 562-563.
doi: 10.2307/2311152. |
| [16] |
J. Milnor, Topology from the Differentiable Viewpoint, University of Virginia Press, 1965.
|
| [17] |
J. Munkres, Topology, 2$^{nd}$ edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
| [18] |
I. Richards,
On the classification of non-compact surfaces, Trans. Am. Math. Soc., 106 (1963), 259-269.
doi: 10.1090/S0002-9947-1963-0143186-0. |
| [19] |
M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, 3$^rd$ edition, Publish or Perish, 1970. |
| [20] |
E. Stein, Complex Analysis, Princeton University Press, Princeton, N.J., 2003.
|
| [21] |
T. W. Tucker,
On the Fox-Artin sphere and surfaces in noncompact 3-manifolds, Q. J. Math., 28 (1977), 243-253.
doi: 10.1093/qmath/28.2.243. |
| [22] |
B. von Kérékjartó,
Über die periodischen transformationen der kreisscheibe und der kugelfläche, Math. Ann., 80 (1919), 36-38.
doi: 10.1007/BF01463232. |
| [23] |
V. B. Yap, Re-imagining the Hardy-Weinberg law, 2013, arXiv: 1307.4417v1. |
show all references
References:
| [1] |
C. Babbage,
An essay towards the calculus of functions, Philos. Trans. Royal Soc., 105 (1815), 389-423.
|
| [2] |
N. Bacaër, A Short History of Mathematical Population Dynamics, Springer-Verlag London Ltd., London, 2011.
doi: 10.1007/978-0-85729-115-8. |
| [3] |
K. Baron and W. Jarczyk,
Recent results on functional equations in a single variable, perspectives and open problems, Aequ. Math., 61 (2001), 1-48.
doi: 10.1007/s000100050159. |
| [4] |
R. H. Bing,
Inequivalent families of periodic homeomorphisms of $E^3$, Ann. of Math., 80 (1964), 78-93.
doi: 10.2307/1970492. |
| [5] |
A. Cima, A. Gasull, F. Mañosas and R. Ortega,
Linearization of planar involutions in $\mathcal C^1$, Ann. Mat. Pura Appl., 194 (2015), 1349-1357.
doi: 10.1007/s10231-014-0423-5. |
| [6] |
A. Cima, A. Gasull, F. Mañosas and R. Ortega,
Smooth linearisation of planar periodic maps, Math. Proc. Camb. Philos. Soc., 167 (2019), 295-320.
doi: 10.1017/S0305004118000336. |
| [7] |
A. Constantin and B. Kolev,
The theorem of Kerérekjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math., 40 (1994), 193-204.
|
| [8] |
G. M. Ewing and W. R. Utz,
Continuous solutions of the functional equation $f^n(x) = f(x)$, Canadian J. Math., 5 (1953), 101-103.
doi: 10.4153/CJM-1953-012-8. |
| [9] |
A. Haefliger,
Plongements différentiables de variétés dans variétés, Comment. Math. Helv., 36 (1962), 47-82.
|
| [10] |
R. Haynes, S. Kwasik, J. Mast and R. Schultz,
Periodic maps on $R^7$ without fixed points, Math. Proc. Camb. Philos. Soc., 132 (2002), 131-136.
doi: 10.1017/S0305004101005345. |
| [11] |
M. Hirsch, Differential Topology, Springer-Verlag, 1976. |
| [12] |
M. Holz, K. Steffens and E. Weitz, Introduction to Cardinal Arithmetic, Birkhäuser Verlag, 2010.
doi: 10.1007/978-3-0346-0330-0. |
| [13] |
G. Ishikawa and T. Nishimura,
Smooth retracts of Euclidean space, Kodai Math. J., 18 (1995), 260-265.
|
| [14] |
W. Jarczyk,
Babbage equation on the circle, Publ. Math., 63 (2003), 389-400.
|
| [15] |
N. McShane,
On the periodicity of homeomorphisms of the real line, Amer. Math. Monthly, 68 (1961), 562-563.
doi: 10.2307/2311152. |
| [16] |
J. Milnor, Topology from the Differentiable Viewpoint, University of Virginia Press, 1965.
|
| [17] |
J. Munkres, Topology, 2$^{nd}$ edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
| [18] |
I. Richards,
On the classification of non-compact surfaces, Trans. Am. Math. Soc., 106 (1963), 259-269.
doi: 10.1090/S0002-9947-1963-0143186-0. |
| [19] |
M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, 3$^rd$ edition, Publish or Perish, 1970. |
| [20] |
E. Stein, Complex Analysis, Princeton University Press, Princeton, N.J., 2003.
|
| [21] |
T. W. Tucker,
On the Fox-Artin sphere and surfaces in noncompact 3-manifolds, Q. J. Math., 28 (1977), 243-253.
doi: 10.1093/qmath/28.2.243. |
| [22] |
B. von Kérékjartó,
Über die periodischen transformationen der kreisscheibe und der kugelfläche, Math. Ann., 80 (1919), 36-38.
doi: 10.1007/BF01463232. |
| [23] |
V. B. Yap, Re-imagining the Hardy-Weinberg law, 2013, arXiv: 1307.4417v1. |
| [1] |
Dyi-Shing Ou, Kenneth James Palmer. A constructive proof of the existence of a semi-conjugacy for a one dimensional map. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 977-992. doi: 10.3934/dcdsb.2012.17.977 |
| [2] |
Xiao Tang, Yingying Zeng, Weinian Zhang. Interval homeomorphic solutions of a functional equation of nonautonomous iterations. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6967-6984. doi: 10.3934/dcds.2020214 |
| [3] |
Chun-Gil Park. Stability of a linear functional equation in Banach modules. Conference Publications, 2003, 2003 (Special) : 694-700. doi: 10.3934/proc.2003.2003.694 |
| [4] |
A. Cibotarica, Jiu Ding, J. Kolibal, Noah H. Rhee. Solutions of the Yang-Baxter matrix equation for an idempotent. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 347-352. doi: 10.3934/naco.2013.3.347 |
| [5] |
Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745 |
| [6] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
| [7] |
Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032 |
| [8] |
Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321 |
| [9] |
Leonid V. Bogachev, Gregory Derfel, Stanislav A. Molchanov. Analysis of the archetypal functional equation in the non-critical case. Conference Publications, 2015, 2015 (special) : 132-141. doi: 10.3934/proc.2015.0132 |
| [10] |
Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 |
| [11] |
Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss, José Valero. Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 439-455. doi: 10.3934/dcdsb.2010.14.439 |
| [12] |
Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001 |
| [13] |
Claire Chavaudret, Stefano Marmi. Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3077-3101. doi: 10.3934/dcds.2022009 |
| [14] |
Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139 |
| [15] |
Mourad Bellassoued, Zouhour Rezig. Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1061-1084. doi: 10.3934/dcdss.2021158 |
| [16] |
Alessandra Celletti, Sara Di Ruzza. Periodic and quasi--periodic orbits of the dissipative standard map. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151 |
| [17] |
Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 |
| [18] |
Wacław Marzantowicz, Piotr Maciej Przygodzki. Finding periodic points of a map by use of a k-adic expansion. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 495-514. doi: 10.3934/dcds.1999.5.495 |
| [19] |
Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 |
| [20] |
Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]