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## Wavelet constructions in non-linear dynamics

### Dorin Ervin Dutkay and Palle E.T. Jorgensen

**Abstract.**
We construct certain Hilbert spaces associated with a class of
non-linear dynamical systems $X$. These are systems which arise from a
generalized self-similarity and an iterated substitution. We show that when
a weight function $W$ on $X$ is given, then we may construct associated Hilbert
spaces $H(W)$ of $L^2$-martingales which have wavelet bases.

*Copyright 2005 American Mathematical Society
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#### Article Info

- ERA Amer. Math. Soc.
**11** (2005), pp. 21-33
- Publisher Identifier: S 1079-6762(05)00143-5
- 2000
*Mathematics Subject Classification*. Primary 60G18; Secondary 42C40, 46G15, 42A65, 28A50, 30D05, 47D07, 37F20
*Key words and phrases.* Measures, projective limits, transfer operator,
martingale, fixed point, multiresolution, Julia set, subshift,
wavelet
- Received by editors October 28, 2004
- Posted on March 7, 2005
- Communicated by Boris Hasselblatt
- Comments (When Available)

**Dorin Ervin Dutkay**

Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419

*E-mail address:* `ddutkay@math.rutgers.edu`

**Palle E.T. Jorgensen**

Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419

*E-mail address:* `jorgen@math.uiowa.edu`

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