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December  2020, 40(12): 6649-6679. doi: 10.3934/dcds.2020131

Generic Birkhoff spectra

Zoltán Buczolich 1,, , Balázs Maga 1, and  Ryo Moore 2,

1. 

Department of Analysis, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117, Budapest, Hungary
2. 

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avda. Vicuña Mackenna 4860, Santiago, Chile

* Corresponding author

Received  May 2019 Published  December 2020 Early access  February 2020

Fund Project: ZB was supported by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124003.
BM was supported by the ÚNKP-18-2 New National Excellence of the Hungarian Ministry of Human Capacities, and by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124749.
RM was partially supported by CONICYT-FONDECYT Postdoctorado 3170279

Suppose that $ \Omega = \{0, 1\}^\mathbb{N} $ and $ \sigma $ is the one-sided shift. The Birkhoff spectrum $ S_{f}(α) = \dim_{H}\Big \{ \omega\in\Omega:\lim\limits_{N \to \infty} \frac{1}{N} \sum\limits_{n = 1}^N f(\sigma^n \omega) = \alpha \Big \}, $ where $ \dim_{H} $ is the Hausdorff dimension. It is well-known that the support of $ S_{f}(α) $ is a bounded and closed interval $ L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*] $ and $ S_{f}(α) $ on $ L_{f} $ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $ f\in C(\Omega) $ in the sense of Baire category. For a dense set in $ C(\Omega) $ the spectrum is not continuous on $ \mathbb{R} $, though for the generic $ f\in C(\Omega) $ the spectrum is continuous on $ \mathbb{R} $, but has infinite one-sided derivatives at the endpoints of $ L_{f} $. We give an example of a function which has continuous $ S_{f} $ on $ \mathbb{R} $, but with finite one-sided derivatives at the endpoints of $ L_{f} $. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions $ f $ and $ g $ are close in $ C(\Omega) $ then $ S_{f} $ and $ S_{g} $ are close on $ L_{f} $ apart from neighborhoods of the endpoints.

Keywords: Birkhoff spectrum, multifractal analysis, Hausdorff dimension, generic/typical continuous functions, symbolic dynamics.
Mathematics Subject Classification: Primary: 37A30; Secondary: 28A80, 37B10, 37C45.
Citation: Zoltán Buczolich, Balázs Maga, Ryo Moore. Generic Birkhoff spectra. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6649-6679. doi: 10.3934/dcds.2020131
References:
[1]

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G. Iommi, T. Jordan and M. Todd, Transience and multifractal analysis, Ann. Inst. Poincaré Anal. Non Linéaire, 34 (2017), 407-421.  doi: 10.1016/j.anihpc.2015.12.007.  Google Scholar

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L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl (9), 82 (2003), 1591-1649.  doi: 10.1016/j.matpur.2003.09.007.  Google Scholar

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Y. Pesin and H. Weiss, The multifractal analysis of Birkhoff averages and large deviations, in Global Analysis of Dynamical Systems, IoP Publishing, Bristol, UK, (2001), 419–431.  Google Scholar

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D. A. Rand, The singularity spectrum f(α) for cookie cutters, Ergodic Theory Dynam. Systems, 9 (1989), 527-541.  doi: 10.1017/S0143385700005162.  Google Scholar

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R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

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J. Schmeling, On the completeness of multifractal spectra, Ergodic Theory Dynam. Systems, 19 (1999), 1595-1616.  doi: 10.1017/S0143385799151988.  Google Scholar

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R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000), 113-143.  doi: 10.1088/0951-7715/13/1/306.  Google Scholar

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F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348.  doi: 10.1017/S0143385702000913.  Google Scholar

show all references

References:
[1]

L. Barreira and B. Saussol, Variational principles and mixed multifractal spectra, Trans. Amer. Math. Soc., 353 (2001), 3919-3944.  doi: 10.1090/S0002-9947-01-02844-6.  Google Scholar

[2]

R. Cawley and R. D. Mauldin, Multifractal decompositions of moran fractals, Adv. Math., 92 (1992), 196-236.  doi: 10.1016/0001-8708(92)90064-R.  Google Scholar

[3]

V. Climenhaga, Multifractal formalism derived from thermodynamics for general dynamical systems, Electron. Res. Announc. Math. Sci., 17 (2010), 1-11.  doi: 10.3934/era.2010.17.1.  Google Scholar

[4]

V. Climenhaga, The thermodynamic approach to multifractal analysis, Ergodic Theory Dynam. Systems, 34 (2014), 1409-1450.  doi: 10.1017/etds.2014.12.  Google Scholar

[5]

H. G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math., Oxford Ser., 20 (1949), 31-36.  doi: 10.1093/qmath/os-20.1.31.  Google Scholar

[6]

A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.  Google Scholar

[7]

D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.  doi: 10.1006/aima.2001.2054.  Google Scholar

[8]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[9]

G. Iommi, T. Jordan and M. Todd, Transience and multifractal analysis, Ann. Inst. Poincaré Anal. Non Linéaire, 34 (2017), 407-421.  doi: 10.1016/j.anihpc.2015.12.007.  Google Scholar

[10]

O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic Theory Dynam. Systems, 39 (2019), 2593-2618.  doi: 10.1017/etds.2017.142.  Google Scholar

[11]

A. Johansson, T. Jordan, A. Öberg and M. Pollicott, Multifractal analysis of non-uniformly hyperbolic systems, Israel J. Math., 177 (2010), 125-144.  doi: 10.1007/s11856-010-0040-y.  Google Scholar

[12]

I. D. Morris, Ergodic optimization for generic continuous functions, Discrete Contin. Dyn. Sys., 27 (2010), 383-388.  doi: 10.3934/dcds.2010.27.383.  Google Scholar

[13]

L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl (9), 82 (2003), 1591-1649.  doi: 10.1016/j.matpur.2003.09.007.  Google Scholar

[14]

Y. Pesin and H. Weiss, The multifractal analysis of Birkhoff averages and large deviations, in Global Analysis of Dynamical Systems, IoP Publishing, Bristol, UK, (2001), 419–431.  Google Scholar

[15]

D. A. Rand, The singularity spectrum f(α) for cookie cutters, Ergodic Theory Dynam. Systems, 9 (1989), 527-541.  doi: 10.1017/S0143385700005162.  Google Scholar

[16]

R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[17]

J. Schmeling, On the completeness of multifractal spectra, Ergodic Theory Dynam. Systems, 19 (1999), 1595-1616.  doi: 10.1017/S0143385799151988.  Google Scholar

[18]

R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000), 113-143.  doi: 10.1088/0951-7715/13/1/306.  Google Scholar

[19]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348.  doi: 10.1017/S0143385702000913.  Google Scholar

Figure 1.  An illustration of Remark 5.2.3
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