Based on model-technical as well as physical considerations a nodal-point relation at bifurcations is proposed for one-dimensional (ID) network morphodynamic models: the ratio between the sediment transports into the downstream branches is proportional to a power of the discharge ratio. The influence of the nodal-point relation on the behaviour of the morphodynamic model is analyzed theoretically. The exponent in the nodal-point relation appears to be crucial for the stability of the bifurcation in the model. For large values of the exponent, the bifurcation is stable, i.e. the downstream branches remain open. For small values of the exponent, the bifurcation is unstable: only one of the branches tends to remain open. The exponent also has a strong influence on the morphological time scales of the network. The conclusions from the analysis have been verified by numerical simulations using a package for one-dimensional network modelling. RÉSUMÉ L'article propose une loi des nueuds aux bifurcations de modèles morphodynamiques filaires de réseaux maillés basée sur des techniques de modélisation aussi bien que sur des considerations physiques : Ie rapport des debits solides dans les branches aval est proportionnel a une puissance du rapport des debits liquides. L'influence de cette loi des nceuds sur Ie comportement du modèle morphodynamique est analysée de facon théorique. L'exposant de la loi des neeuds apparaït comme un element crucial de la stabilité de la bifurcation dans Ie modèle. Pour les grandes valeurs de l'exposant. la bifurcation est stable c'est-a-dire que des branches aval demeurent ouvetes. Pour les petites valeurs de l'exposant, la bifurcation est instable: seulement une des branches tend a demeurer ouverte. L'exposant a également une forte influence sur l'échelle des temps d'évolution morphologique du réseau. Les conclusions de cette analyse ont été vérifiées par des simulations numériques basées sur un code pour réseau filairc.

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Let β > 1 be a non-integer. We consider expansions of the formare generated by means of a Borel map K β defined on {0, 1} N × [0, ⌊β⌋/(β − 1)]. We show existence and uniqueness of an absolutely continuous K β -invariant probability measure w.r.t. mp ⊗ λ, where mp is the Bernoulli measure on {0, 1} N with parameter p (0 < p < 1) and λ is the normalized Lebesgue measure on [0, ⌊β⌋/(β − 1)]. Furthermore, this measure is of the form mp ⊗µ β,p , where µ β,p is equivalent with λ. We establish the fact that the measure of maximal entropy and mp⊗λ are mutually singular. In case the number 1 has a finite greedy expansion with positive coefficients, the measure mp ⊗ µ β,p is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].then the lazy transformation L β : J β → J β is defined by1991 Mathematics Subject Classification. Primary:28D05, Secondary:11K16, 28D20, 37A35, 37A45.

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Let β > 1 be a non-integer. We consider β-expansions of the form ∞ i=1 d i /β i , where the digits (d i ) i≥1 are generated by means of a Borel map K β defined on {0, 1} N ×[0, β /(β − 1)]. We show that K β has a unique mixing measure ν β of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ν β the digits (d i ) i≥1 form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of β-expansions.