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Invariant densities for random systems of the interval

Part of: Ergodic theory Stochastic processes Random dynamical systems Low-dimensional dynamical systems Measure-theoretic ergodic theory

Published online by Cambridge University Press:  29 December 2020

CHARLENE KALLE  and
MARTA MAGGIONI
CHARLENE KALLE*
Affiliation:
Mathematisch Instituut, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands (e-mail:[email protected])
MARTA MAGGIONI
Affiliation:
Mathematisch Instituut, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands (e-mail:[email protected])
*
e-mail:[email protected]
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Abstract

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For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $ -transformations and random Lüroth maps with a hole.

Keywords

absolutely continuous invariant measureβ-expansionsinterval mapLüroth expansionsrandom dynamics

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 28D05: Measure-preserving transformations 37H05: Foundations, general theory of cocycles, algebraic ergodic theory
Secondary: 37A05: Measure-preserving transformations 37A10: One-parameter continuous families of measure-preserving transformations 60G10: Stationary processes
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 141 - 179
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Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

REFERENCES

Abbasi, N., Gharaei, M. and Homburg, A. J.. Iterated function systems of logistic maps: synchronization and intermittency. Nonlinearity 31(8) (2018), 3880–3913.CrossRefGoogle Scholar
Barrionuevo, J., Burton, R. M., Dajani, K. and Kraaikamp, C.. Ergodic properties of generalized Luroth series. TU Delft Report 94–105 (1994), 1–16.Google Scholar
Bahsoun, W. and Góra, P.. Position dependent random maps in one and higher dimensions. Studia Math. 166(3) (2005), 271–286.CrossRefGoogle Scholar
Boyarsky, A. and Góra, P.. Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications). Birkhäuser, Boston, 1997.Google Scholar
Barreira, L. and Iommi, G.. Frequency of digits in the Lüroth expansion. J. Number Theory 129(6) (2009), 1479–1490.CrossRefGoogle Scholar
Buzzi, J.. Absolutely continuous S.R.B. measures for random Lasota–Yorke maps. Trans. Amer. Math. Soc. 352(7) (2000), 3289–3303.CrossRefGoogle Scholar
Dajani, K. and de Vries, M.. Measures of maximal entropy for random $\beta$ -expansions. J. Eur. Math. Soc. 7(1) (2005), 51–68.CrossRefGoogle Scholar
Dajani, K. and de Vries, M.. Invariant densities for random $\beta$ -expansions. J. Eur. Math. Soc. 9(1) (2007), 157–176.CrossRefGoogle Scholar
Dajani, K., Hartono, Y. and Kraaikamp, C.. Mixing properties of $\left(\alpha, \beta \right)$ -expansions. Ergod. Th. & Dynam. Sys. 29(4) (2009), 1119–1140.CrossRefGoogle Scholar
Dajani, K. and Kraaikamp, C.. Random $\beta$ -expansions. Ergod. Th. & Dynam. Sys. 23(2) (2003), 461–479.CrossRefGoogle Scholar
Dajani, K. and Kalle, C.. A note on the greedy $\beta$ -transformation with arbitrary digits. École de Théorie Ergodique (Séminaires et Congrès, 20). Société Mathématique de France, Paris, 2010, pp. 83–104.Google Scholar
Dajani, K. and Kalle, C.. Local dimensions for the random $\beta$ -transformation. New York J. Math. 19 (2013), 285–303.Google Scholar
Erdös, P., Joó, I. and Komornik, V.. Characterization of the unique expansions $1={\sum}_{i=1}^{\infty }{q}^{-{n}_i}$ and related problems. Bull. Soc. Math. France 118(3) (1990), 377–390.CrossRefGoogle Scholar
Eslami, P. and Misiurewicz, M.. Singular limits of absolutely continuous invariant measures for families of transitive maps. J. Difference Equ. Appl. 18(4) (2012), 739–750.CrossRefGoogle Scholar
Fan, A., Liao, L., Ma, J. and Wang, B.. Dimension of Besicovitch–Eggleston sets in countable symbolic space. Nonlinearity 23(5) (2010), 1185–1197.CrossRefGoogle Scholar
Góra, P. and Boyarsky, A.. Absolutely continuous invariant measures for random maps with position dependent probabilities. J. Math. Anal. Appl. 278(1) (2003), 225–242.CrossRefGoogle Scholar
Góra, P., Boyarsky, A. and Islam, S.. Invariant densities of random maps have lower bounds on their supports. J. Appl. Math. Stoch. Anal. 13 (2006), 79175.Google Scholar
Gel’fond, A. O.. A common property of number systems. Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 809–814.Google Scholar
Gui, Y. and Li, W.. Hausdorff dimensions of sets related to Lüroth expansion. Acta Math. Hungar. 150(2) (2016), 286–302.CrossRefGoogle Scholar
Góra, P.. Invariant densities for piecewise linear maps of the unit interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583.CrossRefGoogle Scholar
Inoue, T.. Invariant measures for position dependent random maps with continuous random parameters. Studia Math. 208(1) (2012), 11–29.CrossRefGoogle Scholar
Jager, H. and de Vroedt, C.. Lüroth series and their ergodic properties. Indag. Math. 71(1) (1969), 31–42.CrossRefGoogle Scholar
Keller, G.. Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94(4) (1982), 313–333.CrossRefGoogle Scholar
Kempton, K.. On the invariant density of the random $\beta$ -transformation. Acta Math. Hungar. 142(2) (2014), 403–419.CrossRefGoogle Scholar
Kalpazidou, S., Knopfmacher, A. and Knopfmacher, J.. Lüroth-type alternating series representations for real numbers. Acta Arith. 55(4) (1990), 311–322.CrossRefGoogle Scholar
Kalpazidou, S., Knopfmacher, A. and Knopfmacher, J.. Metric properties of alternating Lüroth series. Port. Math. 48(3) (1991), 319–325.Google Scholar
Kopf, C.. Invariant measures for piecewise linear transformations of the interval. Appl. Math. Comput. 39(2, part II) (1990), 123–144.Google Scholar
Li, Z., Góra, P., Boyarsky, A., Proppe, H. and Eslami, P.. Family of piecewise expanding maps having singular measure as a limit of ACIMs. Ergod. Th. & Dynam. Sys. 33(1) (2013), 158–167.CrossRefGoogle Scholar
Lasota, A. and Mackey, M.C.. Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, 2nd edn (Applied Mathematical Sciences, 97). Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
Lüroth, J.. Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe. Math. Ann. 21(3) (1883), 411–423.CrossRefGoogle Scholar
Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 1974 481–488.CrossRefGoogle Scholar
Li, T. Y. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.CrossRefGoogle Scholar
Morita, T.. Random iteration of one-dimensional transformations. Osaka J. Math. 22(3) (1985), 489–518.Google Scholar
Mance, B. and Tseng, J.. Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists. Acta Arith. 158(1) (2013), 33–47.CrossRefGoogle Scholar
Parry, W.. On the $\beta$ -expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.CrossRefGoogle Scholar
Pelikan, S.. Invariant densities for random maps of the interval. Trans. Amer. Math. Soc. 281(2) (1984), 813–825.CrossRefGoogle Scholar
Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar 8 (1957), 477–493.CrossRefGoogle Scholar
Salát, T.. Zur metrischen Theorie der Lürothschen Entwicklungen der reellen Zahlen. Czechoslovak Math. J. 18 (93) (1968), 489–522.Google Scholar
Shen, L. and Fang, K.. The fractional dimensional theory in Lüroth expansion. Czechoslovak Math. J. 61(136)(3) (2011), 795–807.CrossRefGoogle Scholar
Sidorov, N.. Almost every number has a continuum of $\beta$ -expansions. Amer. Math. Monthly 110(9) (2003), 838–842.Google Scholar
Suzuki, S.. Invariant density functions of random $\beta$ -transformations. Ergod. Th. & Dynam. Sys. 39(4) (2019), 1099–1120.CrossRefGoogle Scholar