We study the dynamics of discrete--time regulatory networks on random digraphs. For this we defin... more We study the dynamics of discrete--time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long--term dynamics of the system. We consider in particular the subnetworks, which we call modules, where the asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks. Additionally we prove that any random initial condition converges almost surely to a periodic attractor.
Journal of Physics A: Mathematical and Theoretical
We describe the transient regime of synchronizing flows defined over a network, introducing a cod... more We describe the transient regime of synchronizing flows defined over a network, introducing a codification of the paths towards synchronization. In all cases, the collection of paths toward synchronization defines a combinatorial structure: the transition diagram. In this paper, besides introducing this general framework, we study in full detail the case of completely connected networks where a monotonic convergence towards synchronization takes place. For these systems we are able to supply a detailed analytical description of the transition diagram describing both, the Laplacian flow and a special case of the Kuramoto model on the same network. This constitutes a first instance of a full symbolic description of the path towards synchronization and illustrates the hidden complexity of the transient regime.
One paradigm of random monoallelic gene expression is that of T-cell receptor (TCR)βallelic exclu... more One paradigm of random monoallelic gene expression is that of T-cell receptor (TCR)βallelic exclusion in T lymphocytes. However, the dynamics that sustain asymmetric choice in TCRβdual allele usage and the production of TCRβmonoallelic expressing T-cells remain poorly understood. Here, we develop a computational model to explore a scheme of TCRβallelic exclusion based on the stochastic initiation of DNA rearrangement [V(D)J recombination] at homologous alleles in T-cell progenitors, and thus account for the genotypic profiles typically associated with allelic exclusion in differentiated T-cells. Disturbances in these dynamics at the level of an individual allele have limited consequences on these pro1les, robust feature of the system that is underscored by our simulations. Our study predicts a biological system in which locus-specific, prime epigenetic allelic activation effects set the stage to both optimize the production of TCRβallelically excluded T-cells and curtail the emergen...
A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps.- Showcase of Blue S... more A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps.- Showcase of Blue Sky Catastrophes.- Directional Metric Entropy and Lyapunov Exponents for Dynamical Systems Generated by Cellular Automata.- On the Complexity of some Geometrical Objects.- Fluctuations of Observables in Dynamical Systems: from Limit Theorems to Concentration Inequalities.- A Theory for Flow Barriers in Discontinuous Dynamical systems.- Nonstandard Analysis of the Behavior of Ergodic Means of Dynamical Systems on Very Big Finite Probability Spaces.- On Measures Resisting Multifractal Analysis.- An Overview of Complex Klenian Groups.-Semigroups of Mappings and Correspondences: Characters and Representations in Holomorphic Dynamical Systems.
Several classical problems in symbolic dynamics concern the characterization of the simplex of me... more Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics—a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.
We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. Thes... more We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes of piecewise contracting maps. Then, we explore the case where the attractor intersects these boundaries by providing examples that show the rich phenomenology of these systems. Due to the discontinuities, the asymptotic behaviour is not always properly represented by the dynamics in the attractor. Hence, we introduce generalized orbits to describe the asymptotic dynamics and its recurrence an...
We consider the full shift T:Ω→Ω where Ω=A^ N, A being a finite alphabet. For a class of potentia... more We consider the full shift T:Ω→Ω where Ω=A^ N, A being a finite alphabet. For a class of potentials which contains in particular potentials ϕ with variation decreasing like O(n^-α) for some α>2, we prove that their corresponding equilibrium state μ_ϕ satisfies a Gaussian concentration bound. Namely, we prove that there exists a constant C>0 such that, for all n and for all separately Lipschitz functions K(x_0,...,x_n-1), the exponential moment of K(x,...,T^n-1x)-∫ K(y,...,T^n-1y) dμ_ϕ(y) is bounded by (C∑_i=0^n-1Lip_i(K)^2). The crucial point is that C is independent of n and K. We then derive various consequences of this inequality. For instance, we obtain bounds on the fluctuations of the empirical frequency of blocks, the speed of convergence of the empirical measure, and speed of Markov approximation of μ_ϕ. We also derive an almost-sure central limit theorem.
Starting from the full--shift on a finite alphabet A, mingling some symbols of A, we obtain a new... more Starting from the full--shift on a finite alphabet A, mingling some symbols of A, we obtain a new full shift on a smaller alphabet B. This amalgamation defines a factor map from (A^ N,T_A) to (B^ N,T_B), where T_A and T_B are the respective shift maps. According to the thermodynamic formalism, to each regular function (`potential') ψ:A^ N→ R, we can associate a unique Gibbs measure μ_ψ. In this article, we prove that, for a large class of potentials, the pushforward measure μ_ψ∘π^-1 is still Gibbsian for a potential ϕ:B^ N→ R having a `bit less' regularity than ψ. In the special case where ψ is a `2--symbol' potential, the Gibbs measure μ_ψ is nothing but a Markov measure and the amalgamation π defines a hidden Markov chain. In this particular case, our theorem can be recast by saying that a hidden Markov chain is a Gibbs measure (for a Hölder potential).
We study the dynamics of discrete--time regulatory networks on random digraphs. For this we defin... more We study the dynamics of discrete--time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long--term dynamics of the system. We consider in particular the subnetworks, which we call modules, where the asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks. Additionally we prove that any random initial condition converges almost surely to a periodic attractor.
Journal of Physics A: Mathematical and Theoretical
We describe the transient regime of synchronizing flows defined over a network, introducing a cod... more We describe the transient regime of synchronizing flows defined over a network, introducing a codification of the paths towards synchronization. In all cases, the collection of paths toward synchronization defines a combinatorial structure: the transition diagram. In this paper, besides introducing this general framework, we study in full detail the case of completely connected networks where a monotonic convergence towards synchronization takes place. For these systems we are able to supply a detailed analytical description of the transition diagram describing both, the Laplacian flow and a special case of the Kuramoto model on the same network. This constitutes a first instance of a full symbolic description of the path towards synchronization and illustrates the hidden complexity of the transient regime.
One paradigm of random monoallelic gene expression is that of T-cell receptor (TCR)βallelic exclu... more One paradigm of random monoallelic gene expression is that of T-cell receptor (TCR)βallelic exclusion in T lymphocytes. However, the dynamics that sustain asymmetric choice in TCRβdual allele usage and the production of TCRβmonoallelic expressing T-cells remain poorly understood. Here, we develop a computational model to explore a scheme of TCRβallelic exclusion based on the stochastic initiation of DNA rearrangement [V(D)J recombination] at homologous alleles in T-cell progenitors, and thus account for the genotypic profiles typically associated with allelic exclusion in differentiated T-cells. Disturbances in these dynamics at the level of an individual allele have limited consequences on these pro1les, robust feature of the system that is underscored by our simulations. Our study predicts a biological system in which locus-specific, prime epigenetic allelic activation effects set the stage to both optimize the production of TCRβallelically excluded T-cells and curtail the emergen...
A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps.- Showcase of Blue S... more A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps.- Showcase of Blue Sky Catastrophes.- Directional Metric Entropy and Lyapunov Exponents for Dynamical Systems Generated by Cellular Automata.- On the Complexity of some Geometrical Objects.- Fluctuations of Observables in Dynamical Systems: from Limit Theorems to Concentration Inequalities.- A Theory for Flow Barriers in Discontinuous Dynamical systems.- Nonstandard Analysis of the Behavior of Ergodic Means of Dynamical Systems on Very Big Finite Probability Spaces.- On Measures Resisting Multifractal Analysis.- An Overview of Complex Klenian Groups.-Semigroups of Mappings and Correspondences: Characters and Representations in Holomorphic Dynamical Systems.
Several classical problems in symbolic dynamics concern the characterization of the simplex of me... more Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics—a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.
We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. Thes... more We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes of piecewise contracting maps. Then, we explore the case where the attractor intersects these boundaries by providing examples that show the rich phenomenology of these systems. Due to the discontinuities, the asymptotic behaviour is not always properly represented by the dynamics in the attractor. Hence, we introduce generalized orbits to describe the asymptotic dynamics and its recurrence an...
We consider the full shift T:Ω→Ω where Ω=A^ N, A being a finite alphabet. For a class of potentia... more We consider the full shift T:Ω→Ω where Ω=A^ N, A being a finite alphabet. For a class of potentials which contains in particular potentials ϕ with variation decreasing like O(n^-α) for some α>2, we prove that their corresponding equilibrium state μ_ϕ satisfies a Gaussian concentration bound. Namely, we prove that there exists a constant C>0 such that, for all n and for all separately Lipschitz functions K(x_0,...,x_n-1), the exponential moment of K(x,...,T^n-1x)-∫ K(y,...,T^n-1y) dμ_ϕ(y) is bounded by (C∑_i=0^n-1Lip_i(K)^2). The crucial point is that C is independent of n and K. We then derive various consequences of this inequality. For instance, we obtain bounds on the fluctuations of the empirical frequency of blocks, the speed of convergence of the empirical measure, and speed of Markov approximation of μ_ϕ. We also derive an almost-sure central limit theorem.
Starting from the full--shift on a finite alphabet A, mingling some symbols of A, we obtain a new... more Starting from the full--shift on a finite alphabet A, mingling some symbols of A, we obtain a new full shift on a smaller alphabet B. This amalgamation defines a factor map from (A^ N,T_A) to (B^ N,T_B), where T_A and T_B are the respective shift maps. According to the thermodynamic formalism, to each regular function (`potential') ψ:A^ N→ R, we can associate a unique Gibbs measure μ_ψ. In this article, we prove that, for a large class of potentials, the pushforward measure μ_ψ∘π^-1 is still Gibbsian for a potential ϕ:B^ N→ R having a `bit less' regularity than ψ. In the special case where ψ is a `2--symbol' potential, the Gibbs measure μ_ψ is nothing but a Markov measure and the amalgamation π defines a hidden Markov chain. In this particular case, our theorem can be recast by saying that a hidden Markov chain is a Gibbs measure (for a Hölder potential).
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Papers by Edgardo Ugalde