Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 12 (2007) > Paper 46 open journal systems 


Random walks on infinite self-similar graphs

Jörg Neunhäuserer, TFH Berlin


Abstract
We introduce a class of rooted infinite self-similar graphs containing the well known Fibonacci graph and graphs associated with Pisot numbers. We consider directed random walks on these graphs and study their entropy and their limit measures. We prove that every infinite self-similar graph has a random walk of full entropy and that the limit measures of this random walks are absolutely continuous.


Full text: PDF

Pages: 1258-1275

Published on: October 15, 2007


Bibliography
  1. Alexander, J. C.; Zagier, Don. The entropy of a certain infinitely convolved Bernoulli measure. J. London Math. Soc. (2) 44 (1991), no. 1, 121--134. MR1122974 (92g:28035)
  2. M.J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J.P. Schreiber, Pisot and Salem numbers, Birkhauser Verlag Basel, (Basel 1992). MR{1187044}
  3. Denker, Manfred; Grillenberger, Christian; Sigmund, Karl. Ergodic theory on compact spaces.Lecture Notes in Mathematics, Vol. 527.Springer-Verlag, Berlin-New York, 1976. iv+360 pp. MR0457675 (56 #15879)
  4. Falconer, Kenneth. Fractal geometry.Mathematical foundations and applications.John Wiley & Sons, Ltd., Chichester, 1990. xxii+288 pp. ISBN: 0-471-92287-0 MR1102677 (92j:28008)
  5. Garsia, Adriano M. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 1962 409--432. MR0137961 (25 #1409)
  6. Grabner, Peter J.; Kirschenhofer, Peter; Tichy, Robert F. Combinatorial and arithmetical properties of linear numeration systems.Special issue: Paul Erdös and his mathematics. Combinatorica 22 (2002), no. 2, 245--267. MR1909085 (2003f:11113)
  7. Katok, Anatole; Hasselblatt, Boris. Introduction to the modern theory of dynamical systems.With a supplementary chapter by Katok and Leonardo Mendoza.Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. xviii+802 pp. ISBN: 0-521-34187-6 MR1326374 (96c:58055)
  8. Krön, B. Growth of self-similar graphs. J. Graph Theory 45 (2004), no. 3, 224--239. MR2037759 (2004m:05090)
  9. Krön, Bernhard; Teufl, Elmar. Asymptotics of the transition probabilities of the simple random walk on self-similar graphs. Trans. Amer. Math. Soc. 356 (2004), no. 1, 393--414 (electronic). MR2020038 (2004k:60130)
  10. Lalley, Steven P. Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution. J. London Math. Soc. (2) 57 (1998), no. 3, 629--654. MR1659849 (99k:11124)
  11. Ledrappier, François; Porzio, Anna. A dimension formula for Bernoulli convolutions. J. Statist. Phys. 76 (1994), no. 5-6, 1307--1327. MR1298104 (95i:58111)
  12. Nicol, Matthew; Sidorov, Nikita; Broomhead, David. On the fine structure of stationary measures in systems which contract-on-average. J. Theoret. Probab. 15 (2002), no. 3, 715--730. MR1922444 (2003i:28008)
  13. Pesin, Yakov B. Dimension theory in dynamical systems.Contemporary views and applications.Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997. xii+304 pp. ISBN: 0-226-66221-7; 0-226-66222-5 MR1489237 (99b:58003)
  14. Sidorov, Nikita; Vershik, Anatoly. Ergodic properties of the Erdös measure, the entropy of the golden shift, and related problems. Monatsh. Math. 126 (1998), no. 3, 215--261. MR1651776 (2000a:28017)
  15. Walters, Peter. An introduction to ergodic theory.Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. ix+250 pp. ISBN: 0-387-90599-5 MR0648108 (84e:28017)
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489