# Period doubling, entropy, and renormalization

Fundamenta Mathematicae (1998)

- Volume: 155, Issue: 3, page 237-249
- ISSN: 0016-2736

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topHu, Jun, and Tresser, Charles. "Period doubling, entropy, and renormalization." Fundamenta Mathematicae 155.3 (1998): 237-249. <http://eudml.org/doc/212254>.

@article{Hu1998,

abstract = {We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point. Similar techniques then allow us to show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable with the diameters of all periodic intervals going to zero as the period goes to infinity.},

author = {Hu, Jun, Tresser, Charles},

journal = {Fundamenta Mathematicae},

keywords = {renormalization; topological entropy; sawtooth maps; multimodal maps},

language = {eng},

number = {3},

pages = {237-249},

title = {Period doubling, entropy, and renormalization},

url = {http://eudml.org/doc/212254},

volume = {155},

year = {1998},

}

TY - JOUR

AU - Hu, Jun

AU - Tresser, Charles

TI - Period doubling, entropy, and renormalization

JO - Fundamenta Mathematicae

PY - 1998

VL - 155

IS - 3

SP - 237

EP - 249

AB - We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point. Similar techniques then allow us to show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable with the diameters of all periodic intervals going to zero as the period goes to infinity.

LA - eng

KW - renormalization; topological entropy; sawtooth maps; multimodal maps

UR - http://eudml.org/doc/212254

ER -

## References

top- [AKM] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. Zbl0127.13102
- [BORT] H. Bass, M. V. Otero-Espinar, D. N. Rockmore and C. P. L. Tresser, Cyclic Renormalization and Automorphism Groups of Rooted Trees, Lecture Notes in Math. 1621, Springer, Berlin, 1996. Zbl0847.58019
- [Bl1] L. Block, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978), 576-580. Zbl0365.58015
- [Bl2] L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391-398. Zbl0386.54025
- [BC] L. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992.
- [BH] L. Block and D. Hart, The bifurcation of periodic orbits of one-dimensional maps, Ergodic Theory Dynam. Systems 2 (1982), 125-129. Zbl0507.58035
- [Bo] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414. Zbl0212.29201
- [BF] R. Bowen and J. Franks, The periodic points of the disc and the interval, Topology 15 (1976), 337-342. Zbl0346.58010
- [DGMT] S. P. Dawson, R. Galeeva, J. Milnor and C. Tresser, A monotonicity conjecture for real cubic maps, in: Real and Complex Dynamical Systems, B. Branner and P. Hjorth (eds.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 464, Kluwer, Dordrecht, 1995, 165-183. Zbl0909.54013
- [Ge] T. Gedeon, Stable and non-stable non-chaotic maps of the interval, Math. Slovaca 41 (1991), 379-391. Zbl0762.58014
- [Hu] J. Hu, Renormalization, rigidity and universality in bifurcation theory, Ph.D. dissertation, Department of Math., City Univ. of New York, 1995.
- [HS] J. Hu and D. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems 17 (1997), 173-186. Zbl0997.37010
- [JS1] V. Jiménez López and L. Snoha, There are no piecewise linear maps of type ${2}^{\infty}$, preprint, 1994.
- [JS2] V. Jiménez López and L. Snoha, All maps of type ${2}^{\infty}$ are boundary maps, preprint. Zbl0877.26005
- [Kl] P. E. Kloeden, Chaotic difference equations are dense, Bull. Austral. Math. Soc. 15 (1976), 371-379. Zbl0335.39001
- [La] O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjecture, Bull. Amer. Math. Soc. (N.S.) 6 (1984), 427-434.
- [MMS] M. Martens, W. de Melo and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), 273-318. Zbl0761.58007
- [MiT] J. Milnor and W. Thurston, On iterated maps of the interval, in: Lecture Notes in Math. 1342, Springer, Berlin, 1988, 465-563.
- [MiTr] J. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, to appear.
- [M1] M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 167-169. Zbl0459.54031
- [M2] M. Misiurewicz, Invariant measures for continuous transformations of [0,1] with zero topological entropy, in: Lecture Notes in Math. 729, Springer, Berlin, 1979, 144-152.
- [OT] M. V. Otero-Espinar and C. Tresser, Global complexity and essential simplicity: A conjectural picture of the boundary of chaos for smooth endomorphisms of the interval, Phys. D 39 (1989), 163-168. Zbl0696.58033
- [Sm] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269-282. Zbl0639.54029
- [Mo] F. J. Soares Moreira, Applications du disque infiniment renormalisables, Ph.D. thesis, Nice, 1997.
- [Su] D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, in: Mathematics into the Twenty-first Century, American Mathematical Society Centennial Publications, Vol. II, Amer. Math. Soc., Providence, R.I., 1992, 417-466.
- [Ya] K. Yano, A remark on the topological entropy of homeomorphisms, Invent. Math. 59 (1980), 215-220. Zbl0434.54010