December  2021, 41(12): 5633-5658. doi: 10.3934/dcds.2021091

Period tripling and quintupling renormalizations below $ C^2 $ space

Department of Mathematics, Indian Institute of Technology Jodhpur, Rajasthan, India-342037, India

* Corresponding author: V.V.M.S. Chandramouli

Received  October 2020 Revised  February 2021 Published  December 2021 Early access  June 2021

In this paper, we explore the period tripling and period quintupling renormalizations below $ C^2 $ class of unimodal maps. We show that for a given proper scaling data there exists a renormalization fixed point on the space of piece-wise affine maps which are infinitely renormalizable. Furthermore, we show that this renormalization fixed point is extended to a $ C^{1+Lip} $ unimodal map, considering the period tripling and period quintupling combinatorics. Moreover, we show that there exists a continuum of fixed points of renormalizations by considering a small variation on the scaling data. Finally, this leads to the fact that the tripling and quintupling renormalizations acting on the space of $ C^{1+Lip} $ unimodal maps have unbounded topological entropy.

Citation: Rohit Kumar, V.V.M.S. Chandramouli. Period tripling and quintupling renormalizations below $ C^2 $ space. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5633-5658. doi: 10.3934/dcds.2021091
References:
[1]

G. BirkhoffM. Martens and C. Tresser, On the scaling structure for period doubling, Astérisque, 286 (2003), 167-186.   Google Scholar

[2]

V. V. M. S. ChandramouliM. MartensW. D. Melo and C. P. Tresser, Chaotic period doubling, Ergodic Theory and Dynamical Systems, 29 (2009), 381-418.  doi: 10.1017/S0143385708080371.  Google Scholar

[3]

P. Coullet and C. Tresser, Itération d'endomorphisms et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), 577-580.   Google Scholar

[4]

A. M. Davie, Period doubling for $C^{2+\epsilon}$ mappings, Commun. Math. Phys., 176 (1996), 261-272.  doi: 10.1007/BF02099549.  Google Scholar

[5]

E. D. FariaW. D. Melo and A. Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings, Ann. of Math., 164 (2006), 731-824.  doi: 10.4007/annals.2006.164.731.  Google Scholar

[6]

M. J. Feigenbaum, Quantitative universality for a class of non-linear transformations, J. Stat. Phys., 19 (1978), 25-52.  doi: 10.1007/BF01020332.  Google Scholar

[7]

M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Stat. Phys., 21 (1979), 669-706.  doi: 10.1007/BF01107909.  Google Scholar

[8]

O. Lanford-III, A computer assisted proof of the Feigenbaum conjecture, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[9]

M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture, Ann. of Math., 149 (1999), 319-420.  doi: 10.2307/120968.  Google Scholar

[10]

W. D. Melo and S. V. Strien, One-Dimensional Dynamics, 1$^{st}$ edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[11]

C. Tresser, Fine Structure of Universal Cantor Sets, in Instabilities and Nonequilibrium Structures III, E. Tirapegui and W. Zeller Eds., (Kluwer, Dordrecht/Boston/London), 64 (1991), 27-42.  Google Scholar

show all references

References:
[1]

G. BirkhoffM. Martens and C. Tresser, On the scaling structure for period doubling, Astérisque, 286 (2003), 167-186.   Google Scholar

[2]

V. V. M. S. ChandramouliM. MartensW. D. Melo and C. P. Tresser, Chaotic period doubling, Ergodic Theory and Dynamical Systems, 29 (2009), 381-418.  doi: 10.1017/S0143385708080371.  Google Scholar

[3]

P. Coullet and C. Tresser, Itération d'endomorphisms et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), 577-580.   Google Scholar

[4]

A. M. Davie, Period doubling for $C^{2+\epsilon}$ mappings, Commun. Math. Phys., 176 (1996), 261-272.  doi: 10.1007/BF02099549.  Google Scholar

[5]

E. D. FariaW. D. Melo and A. Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings, Ann. of Math., 164 (2006), 731-824.  doi: 10.4007/annals.2006.164.731.  Google Scholar

[6]

M. J. Feigenbaum, Quantitative universality for a class of non-linear transformations, J. Stat. Phys., 19 (1978), 25-52.  doi: 10.1007/BF01020332.  Google Scholar

[7]

M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Stat. Phys., 21 (1979), 669-706.  doi: 10.1007/BF01107909.  Google Scholar

[8]

O. Lanford-III, A computer assisted proof of the Feigenbaum conjecture, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[9]

M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture, Ann. of Math., 149 (1999), 319-420.  doi: 10.2307/120968.  Google Scholar

[10]

W. D. Melo and S. V. Strien, One-Dimensional Dynamics, 1$^{st}$ edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[11]

C. Tresser, Fine Structure of Universal Cantor Sets, in Instabilities and Nonequilibrium Structures III, E. Tirapegui and W. Zeller Eds., (Kluwer, Dordrecht/Boston/London), 64 (1991), 27-42.  Google Scholar

Figure 1.  Intervals of next generations
Figure 2.  Period triple interval combinatorics ($ I_2^n \rightarrow I_3^n \rightarrow I_1^n \rightarrow I_2^n) $
Figure 4.  Period three cobweb diagram
Figure 5.  Length of the intervals gaps
Figure 6.  (a), (b), (c) and (d) show the graphs of $ S_1(c) $, $ S_2(c) $, $ S_3(c) $ and $ (S_1+S_2+S_3)(c) $ respectively
Figure 7.  The graph of $ \mathcal{R} : F_d \rightarrow \mathbb{R} $ and the diagonal $ \mathcal{R}(c) = c $
Figure 8.  The graph of map $ f_{s^*} $
Figure 9.  Extension of $ f_{s^*} $
Figure 10.  $ \epsilon $- perturbation of $ u_c^3(0) $
Figure 11.  (a): $ \mathcal{R}(c, \epsilon) $ and $ \mathcal{R}(c) $ for $ \epsilon < 1 $ and (b): $ \mathcal{R}(c, \epsilon) $ and $ \mathcal{R}(c) $ for $ \epsilon > 1 $
Figure 14.  The intervals of next generations
Figure 16.  Period quintuple combinatorics ($ I_2^n \rightarrow I_5^n \rightarrow I_1^n \rightarrow I_3^n \rightarrow I_4^n \rightarrow I_2^n) $
Figure 17.  Length of the intervals and gaps
Figure 18.  The graphs of $ S_1(c), $ $ S_2(c), $ $ S_3(c), $ $ S_4(c) $ and $ S_5(c) $
Figure 19.  The graph of $ \mathcal{R} : f_d \rightarrow \mathbb{R} $ and the diagonal $ \mathcal{R}(c) = c $
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