Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T04:12:19.513Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant density functions of random $\unicode[STIX]{x1D6FD}$-transformations

Published online by Cambridge University Press:  07 September 2017

SHINTARO SUZUKI
SHINTARO SUZUKI*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan email si-suzuki@cr.math.sci.osaka-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the random $\unicode[STIX]{x1D6FD}$-transformation $K_{\unicode[STIX]{x1D6FD}}$ introduced by Dajani and Kraaikamp [Random $\unicode[STIX]{x1D6FD}$-expansions. Ergod. Th. & Dynam. Sys.23 (2003), 461–479], which is defined on $\{0,1\}^{\mathbb{N}}\times [0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We give an explicit formula for the density function of a unique $K_{\unicode[STIX]{x1D6FD}}$-invariant probability measure absolutely continuous with respect to the product measure $m_{p}\otimes \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$, where $m_{p}$ is the $(1-p,p)$-Bernoulli measure on $\{0,1\}^{\mathbb{N}}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$ is the normalized Lebesgue measure on $[0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We apply the explicit formula for the density function to evaluate its upper and lower bounds and to investigate its continuity as a function of the two parameters $p$ and $\unicode[STIX]{x1D6FD}$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 1099 - 1120
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boyarsky, A. and Góra, P.. Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications) . Birkhäuser, Boston, MA, 1997.Google Scholar
Buzzi, J.. Absolutely continuous SRB measures for random Lasota–Yorke maps. Trans. Amer. Math. Soc. 352 (2000), 3289–3303.Google Scholar
Dajani, K. and Kraaikamp, C.. From greedy to lazy expansions and their driving dynamics. Expo. Math. 20 (2002), 315–327.Google Scholar
Dajani, K. and Kraaikamp, C.. Random 𝛽-expansions. Ergod. Th. & Dynam. Sys. 23 (2003), 461–479.Google Scholar
Dajani, K. and de Vries, M.. Measures of maximal entropy for random 𝛽-expansions. J. Eur. Math. Soc. 7 (2005), 51–68.Google Scholar
Dajani, K. and de Vries, M.. Invariant densities for random 𝛽-expansions. J. Eur. Math. Soc. 9 (2007), 157–176.Google Scholar
Erdös, P., Joó, I. and Komornik, V.. Characterization of the unique expansions 1 =∑ i=1 ∞ q -n i and related problems. Bull. Soc. Math. France 118 (1990), 377–390.Google Scholar
Gel’fond, A.. A common property of number systems. Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 809–814.Google Scholar
Kempton, T.. On the invariant density of the random 𝛽-transformation. Acta Math. Hungar. 142 (2014), 403–419.Google Scholar
Lasota, A. and Yorke, J.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.Google Scholar
Li, T. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.Google Scholar
Liao, L. and Steiner, W.. Dynamical properties of the negative beta transformation. Ergod. Th. & Dynam. Sys. 32 (2012), 1673–1690.Google Scholar
Morita, T.. Random iteration of one-dimensional transformations. Osaka J. Math. 22 (1985), 489–518.Google Scholar
Morita, T.. Deterministic version lemmas in ergodic theory of random dynamical systems. Hiroshima Math. J. 18 (1988), 15–29.Google Scholar
Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
Pelikan, S.. Invariant densities for random maps of the interval. Trans. Amer. Math. Soc. 281 (1984), 813–825.Google Scholar
Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.Google Scholar
Sidorov, N.. Almost every number has a continuum of 𝛽-expansions. Amer. Math. Monthly 110 (2003), 838–842.Google Scholar
Thompson, D.. Generalized beta-transformations and the entropy of unimodal maps. Comment. Math. Helv. Preprint, 2016, arXiv:1602.03518, to appear.Google Scholar