---
_id: '4200'
abstract:
- lang: eng
text: Zebrafish embryos homozygous for the masterblind (mb1) mutation exhibit a
striking phenotype in which the eyes and telencephalon are reduced or absent and
diencephalic fates expand to the front of the brain. Here we show that mb1(-/-)
embryos carry an amino-acid change at a conserved site in the Wnt pathway scaffolding
protein, Axin1. The amino-acid substitution present in the mbl allele abolishes
the binding of Axin to Gsk3 and affects Tcf-dependent transcription. Therefore,
Gsk3 activity may be decreased in mbl(-/-) embryos and in support of this possibility,
overexpression of either wild-type Axin1 or Gsk3 beta can restore eye and telencephalic
fates to mb1(-/-) embryos. Our data reveal a crucial role for Axin1-dependent
inhibition of the Wnt pathway in the early regional subdivision of the anterior
neural plate into telencephalic, diencephalic, and eye-forming territories.
article_processing_charge: No
author:
- first_name: Carl-Philipp J
full_name: Heisenberg, Carl-Philipp J
id: 39427864-F248-11E8-B48F-1D18A9856A87
last_name: Heisenberg
orcid: 0000-0002-0912-4566
- first_name: Corinne
full_name: Houart, Corinne
last_name: Houart
- first_name: Masaya
full_name: Take Uchi, Masaya
last_name: Take Uchi
- first_name: Gerd
full_name: Rauch, Gerd
last_name: Rauch
- first_name: Neville
full_name: Young, Neville
last_name: Young
- first_name: Pedro
full_name: Coutinho, Pedro
last_name: Coutinho
- first_name: Ichiro
full_name: Masai, Ichiro
last_name: Masai
- first_name: Luca
full_name: Caneparo, Luca
last_name: Caneparo
- first_name: Miguel
full_name: Concha, Miguel
last_name: Concha
- first_name: Robert
full_name: Geisler, Robert
last_name: Geisler
- first_name: Trevor
full_name: Dale, Trevor
last_name: Dale
- first_name: Stephen
full_name: Wilson, Stephen
last_name: Wilson
- first_name: Derek
full_name: Stemple, Derek
last_name: Stemple
citation:
ama: Heisenberg C-PJ, Houart C, Take Uchi M, et al. A mutation in the Gsk3-binding
domain of zebrafish Masterblind/Axin1 leads to a fate transformation of telencephalon
and eyes to diencephalon. *Genes and Development*. 2001;15(11):1427-1434.
doi:10.1101/gad.194301
apa: Heisenberg, C.-P. J., Houart, C., Take Uchi, M., Rauch, G., Young, N., Coutinho,
P., … Stemple, D. (2001). A mutation in the Gsk3-binding domain of zebrafish Masterblind/Axin1
leads to a fate transformation of telencephalon and eyes to diencephalon. *Genes
and Development*, *15*(11), 1427–1434. https://doi.org/10.1101/gad.194301
chicago: 'Heisenberg, Carl-Philipp J, Corinne Houart, Masaya Take Uchi, Gerd Rauch,
Neville Young, Pedro Coutinho, Ichiro Masai, et al. “A Mutation in the Gsk3-Binding
Domain of Zebrafish Masterblind/Axin1 Leads to a Fate Transformation of Telencephalon
and Eyes to Diencephalon.” *Genes and Development* 15, no. 11 (2001): 1427–34.
https://doi.org/10.1101/gad.194301.'
ieee: C.-P. J. Heisenberg *et al.*, “A mutation in the Gsk3-binding domain
of zebrafish Masterblind/Axin1 leads to a fate transformation of telencephalon
and eyes to diencephalon,” *Genes and Development*, vol. 15, no. 11, pp.
1427–1434, 2001.
ista: Heisenberg C-PJ, Houart C, Take Uchi M, Rauch G, Young N, Coutinho P, Masai
I, Caneparo L, Concha M, Geisler R, Dale T, Wilson S, Stemple D. 2001. A mutation
in the Gsk3-binding domain of zebrafish Masterblind/Axin1 leads to a fate transformation
of telencephalon and eyes to diencephalon. Genes and Development. 15(11), 1427–1434.
mla: Heisenberg, Carl-Philipp J., et al. “A Mutation in the Gsk3-Binding Domain
of Zebrafish Masterblind/Axin1 Leads to a Fate Transformation of Telencephalon
and Eyes to Diencephalon.” *Genes and Development*, vol. 15, no. 11, Cold
Spring Harbor Laboratory Press, 2001, pp. 1427–34, doi:10.1101/gad.194301.
short: C.-P.J. Heisenberg, C. Houart, M. Take Uchi, G. Rauch, N. Young, P. Coutinho,
I. Masai, L. Caneparo, M. Concha, R. Geisler, T. Dale, S. Wilson, D. Stemple,
Genes and Development 15 (2001) 1427–1434.
date_created: 2018-12-11T12:07:33Z
date_published: 2001-06-01T00:00:00Z
date_updated: 2020-10-07T07:09:14Z
day: '01'
doi: 10.1101/gad.194301
extern: '1'
intvolume: ' 15'
issue: '11'
language:
- iso: eng
month: '06'
oa_version: None
page: 1427 - 1434
publication: Genes and Development
publication_status: published
publisher: Cold Spring Harbor Laboratory Press
publist_id: '1916'
status: public
title: A mutation in the Gsk3-binding domain of zebrafish Masterblind/Axin1 leads
to a fate transformation of telencephalon and eyes to diencephalon
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 15
year: '2001'
...
---
_id: '8522'
abstract:
- lang: eng
text: For diffeomorphisms of smooth compact manifolds, we consider the problem of
how fast the number of periodic points with period $n$grows as a function of $n$.
In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily
fast growth is possible; in fact, the first author has shown that arbitrarily
fast growth is topologically (Baire) generic for $C^2$ or smoother diffeomorphisms.
In the present work we show that, by contrast, for a measure-theoretic notion
of genericity we call ``prevalence'', the growth is not much faster than exponential.
Specifically, we show that for each $\delta > 0$, there is a prevalent set of
( $C^{1+\rho}$ or smoother) diffeomorphisms for which the number of period $n$
points is bounded above by $\operatorname{exp}(C n^{1+\delta})$ for some $C$ independent
of $n$. We also obtain a related bound on the decay of the hyperbolicity of the
periodic points as a function of $n$. The contrast between topologically generic
and measure-theoretically generic behavior for the growth of the number of periodic
points and the decay of their hyperbolicity shows this to be a subtle and complex
phenomenon, reminiscent of KAM theory.
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
- first_name: Brian R.
full_name: Hunt, Brian R.
last_name: Hunt
citation:
ama: Kaloshin V, Hunt BR. A stretched exponential bound on the rate of growth of
the number of periodic points for prevalent diffeomorphisms I. *Electronic Research
Announcements of the American Mathematical Society*. 2001;7(4):17-27. doi:10.1090/s1079-6762-01-00090-7
apa: Kaloshin, V., & Hunt, B. R. (2001). A stretched exponential bound on the
rate of growth of the number of periodic points for prevalent diffeomorphisms
I. *Electronic Research Announcements of the American Mathematical Society*,
*7*(4), 17–27. https://doi.org/10.1090/s1079-6762-01-00090-7
chicago: 'Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on
the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms
I.” *Electronic Research Announcements of the American Mathematical Society*
7, no. 4 (2001): 17–27. https://doi.org/10.1090/s1079-6762-01-00090-7.'
ieee: V. Kaloshin and B. R. Hunt, “A stretched exponential bound on the rate of
growth of the number of periodic points for prevalent diffeomorphisms I,” *Electronic
Research Announcements of the American Mathematical Society*, vol. 7, no. 4,
pp. 17–27, 2001.
ista: Kaloshin V, Hunt BR. 2001. A stretched exponential bound on the rate of growth
of the number of periodic points for prevalent diffeomorphisms I. Electronic Research
Announcements of the American Mathematical Society. 7(4), 17–27.
mla: Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate
of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms I.” *Electronic
Research Announcements of the American Mathematical Society*, vol. 7, no. 4,
American Mathematical Society, 2001, pp. 17–27, doi:10.1090/s1079-6762-01-00090-7.
short: V. Kaloshin, B.R. Hunt, Electronic Research Announcements of the American
Mathematical Society 7 (2001) 17–27.
date_created: 2020-09-18T10:49:56Z
date_published: 2001-04-18T00:00:00Z
date_updated: 2020-10-12T13:26:39Z
day: '18'
doi: 10.1090/s1079-6762-01-00090-7
extern: '1'
intvolume: ' 7'
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
month: '04'
oa_version: None
page: 17-27
publication: Electronic Research Announcements of the American Mathematical Society
publication_identifier:
issn:
- 1079-6762
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: A stretched exponential bound on the rate of growth of the number of periodic
points for prevalent diffeomorphisms I
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2001'
...
---
_id: '8524'
abstract:
- lang: eng
text: 'A number α∈R is diophantine if it is not well approximable by rationals,
i.e. for some C,ε>0 and any relatively prime p,q∈Z we have |αq−p|>Cq−1−ε. It is
well-known and is easy to prove that almost every α in R is diophantine. In this
paper we address a noncommutative version of the diophantine properties. Consider
a pair A,B∈SO(3) and for each n∈Z+ take all possible words in A, A -1, B, and
B - 1 of length n, i.e. for a multiindex I=(i1,i1,…,im,jm) define |I|=∑mk=1(|ik|+|jk|)=n
and \( W_n(A,B ) = \{W_{\cal I}(A,B) = A^{i_1} B^{j_1} \dots A^{i_m} B^{j_m}\}_{|{\cal
I|}=n \).¶Gamburd—Jakobson—Sarnak [GJS] raised the problem: prove that for Haar
almost every pair A,B∈SO(3) the closest distance of words of length n to the identity,
i.e. sA,B(n)=min|I|=n∥WI(A,B)−E∥, is bounded from below by an exponential function
in n. This is the analog of the diophantine property for elements of SO(3). In
this paper we prove that s A,B (n) is bounded from below by an exponential function
in n 2. We also exhibit obstructions to a “simple” proof of the exponential estimate
in n.'
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
- first_name: I.
full_name: Rodnianski, I.
last_name: Rodnianski
citation:
ama: Kaloshin V, Rodnianski I. Diophantine properties of elements of SO(3). *Geometric
And Functional Analysis*. 2001;11(5):953-970. doi:10.1007/s00039-001-8222-8
apa: Kaloshin, V., & Rodnianski, I. (2001). Diophantine properties of elements
of SO(3). *Geometric And Functional Analysis*, *11*(5), 953–970. https://doi.org/10.1007/s00039-001-8222-8
chicago: 'Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements
of SO(3).” *Geometric And Functional Analysis* 11, no. 5 (2001): 953–70.
https://doi.org/10.1007/s00039-001-8222-8.'
ieee: V. Kaloshin and I. Rodnianski, “Diophantine properties of elements of SO(3),”
*Geometric And Functional Analysis*, vol. 11, no. 5, pp. 953–970, 2001.
ista: Kaloshin V, Rodnianski I. 2001. Diophantine properties of elements of SO(3).
Geometric And Functional Analysis. 11(5), 953–970.
mla: Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements of
SO(3).” *Geometric And Functional Analysis*, vol. 11, no. 5, Springer Nature,
2001, pp. 953–70, doi:10.1007/s00039-001-8222-8.
short: V. Kaloshin, I. Rodnianski, Geometric And Functional Analysis 11 (2001) 953–970.
date_created: 2020-09-18T10:50:11Z
date_published: 2001-12-01T00:00:00Z
date_updated: 2020-10-12T13:24:12Z
day: '01'
doi: 10.1007/s00039-001-8222-8
extern: '1'
intvolume: ' 11'
issue: '5'
language:
- iso: eng
month: '12'
oa_version: None
page: 953-970
publication: Geometric And Functional Analysis
publication_identifier:
issn:
- 1016-443X
- 1420-8970
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Diophantine properties of elements of SO(3)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 11
year: '2001'
...
---
_id: '8521'
abstract:
- lang: eng
text: We continue the previous article's discussion of bounds, for prevalent diffeomorphisms
of smooth compact manifolds, on the growth of the number of periodic points and
the decay of their hyperbolicity as a function of their period $n$. In that article
we reduced the main results to a problem, for certain families of diffeomorphisms,
of bounding the measure of parameter values for which the diffeomorphism has (for
a given period $n$) an almost periodic point that is almost nonhyperbolic. We
also formulated our results for $1$-dimensional endomorphisms on a compact interval.
In this article we describe some of the main techniques involved and outline the
rest of the proof. To simplify notation, we concentrate primarily on the $1$-dimensional
case.
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
- first_name: Brian R.
full_name: Hunt, Brian R.
last_name: Hunt
citation:
ama: Kaloshin V, Hunt BR. A stretched exponential bound on the rate of growth of
the number of periodic points for prevalent diffeomorphisms II. *Electronic
Research Announcements of the American Mathematical Society*. 2001;7(5):28-36.
doi:10.1090/s1079-6762-01-00091-9
apa: Kaloshin, V., & Hunt, B. R. (2001). A stretched exponential bound on the
rate of growth of the number of periodic points for prevalent diffeomorphisms
II. *Electronic Research Announcements of the American Mathematical Society*,
*7*(5), 28–36. https://doi.org/10.1090/s1079-6762-01-00091-9
chicago: 'Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on
the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms
II.” *Electronic Research Announcements of the American Mathematical Society*
7, no. 5 (2001): 28–36. https://doi.org/10.1090/s1079-6762-01-00091-9.'
ieee: V. Kaloshin and B. R. Hunt, “A stretched exponential bound on the rate of
growth of the number of periodic points for prevalent diffeomorphisms II,” *Electronic
Research Announcements of the American Mathematical Society*, vol. 7, no. 5,
pp. 28–36, 2001.
ista: Kaloshin V, Hunt BR. 2001. A stretched exponential bound on the rate of growth
of the number of periodic points for prevalent diffeomorphisms II. Electronic
Research Announcements of the American Mathematical Society. 7(5), 28–36.
mla: Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate
of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms II.”
*Electronic Research Announcements of the American Mathematical Society*,
vol. 7, no. 5, American Mathematical Society, 2001, pp. 28–36, doi:10.1090/s1079-6762-01-00091-9.
short: V. Kaloshin, B.R. Hunt, Electronic Research Announcements of the American
Mathematical Society 7 (2001) 28–36.
date_created: 2020-09-18T10:49:43Z
date_published: 2001-04-24T00:00:00Z
date_updated: 2020-10-12T13:28:17Z
day: '24'
doi: 10.1090/s1079-6762-01-00091-9
extern: '1'
intvolume: ' 7'
issue: '5'
keyword:
- General Mathematics
language:
- iso: eng
month: '04'
oa_version: None
page: 28-36
publication: Electronic Research Announcements of the American Mathematical Society
publication_identifier:
issn:
- 1079-6762
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: A stretched exponential bound on the rate of growth of the number of periodic
points for prevalent diffeomorphisms II
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2001'
...