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Self-similarity of the corrections to
the ergodic theorem for the
Pascal-adic transformation
Elise Janvresse, Thierry de la Rue, Yvan Velenik
Laboratoire de Mathematiques Raphael Salem
CENTRE NATIONAL
DE LA RECHERCHE
SCIENTIFIQUE
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformationThe Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
1 The Pascal-adic transformation
2 Self-similar structure of the basic blocks
3 Ergodic interpretation
4 Generalizations and related problems
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformationThe Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Pascal Graph
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformationThe Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Pascal Graph
1
2
n
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformationThe Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Pascal Graph
1
2
n
(n,0) (n,k) (n,n)
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformationThe Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Pascal Graph
1
2
n
(n,0) (n,k) (n,n)
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformationThe Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Pascal Graph
0 1x=01100100111...
1
2
n
(n,0) (n,k) (n,n)
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformation(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)
The Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Recursive enumeration of trajectories
We list all trajectories
going through (n,k)
and xed beyond
this point.
(n,k)
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformation(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)
The Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Recursive enumeration of trajectories
Firstthosecomingfrom
(n 1,k 1),
(n,k)
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformation(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)(n,k)
The Pascal-adic transformation
Introduction to the transformation
Self-similar structure of the basic blocks
Invariant measures
Ergodic interpretation
Coding: basic blocks
Generalizations and related problems
Recursive enumeration of trajectories
Firstthosecomingfrom
(n 1,k 1),
(n,k)
T. de la Rue, E. Janvresse, Y. Velenik Self-similarity of the Pascal-adic transformation