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

Received  January 2020 Revised  June 2020 Published  February 2021 Early access  August 2020

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 $.

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. CimaA. GasullF. 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. CimaA. GasullF. 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. HaynesS. KwasikJ. 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. CimaA. GasullF. 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. CimaA. GasullF. 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. HaynesS. KwasikJ. 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.

Figure 1.  Graph of a generic idempotent continuous function in $ \mathbb R $
Figure 2.  Graph of the function $ f_\lambda $ with $ \lambda = (0.11001\dots)_2 $
Figure 3.  Graph of the function $ f_\lambda $ with $ \lambda = (0.11001\dots)_2 $
Figure 4.  Concatenation of two overhand knots, one figure eight knot and one overhand knot
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