R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshif... more 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, 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 to expose some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way of the possibilities of this correspondence. 1 σ-invariant means that σzX " X for all z P Z 2 R
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...
It is a current practice to organize biological data in a network structure where vertices repres... more It is a current practice to organize biological data in a network structure where vertices represent biological components and arrows represent their interactions. A great diversity of graph theoretical notions, such as clustering coefficient, network motifs, centrality, degree distribution, etc., have been developed in order to characterize the structure of these networks. However, none of the existent characterizations allow us to determine global similarity among networks of different sizes. It is the aim of the present paper to introduce a mathematical tool to compare networks not only with regard to their topological structure, but also in their dynamical capabilities. For this reason we aim to propose a pseudo-distance between networks, built around the notions of determination and dominancy, concepts recently introduced in the context of regulatory dynamics on networks. We use our proposed pseudo-distance to compare networks from the following bacteria: E. coli, B. subtilis, P. aeruginosa, M. tuberculosis, S. aureus and C. glutamicum. We also use this pseudo-distance to compare these real bacterial networks with equivalent homogeneous, scale-free and geometric three dimensional random networks. We found that even when bacterial networks are characterized with different levels of detail, have different sizes and represent different aspects of the organisms, the proposed pseudo-distance captures all these characteristics, and indicates how similar they are or not from random networks.
International Journal of Society Systems Science, 2014
Spatial structure of cities is the substratum over which human urban life develops. Understanding... more Spatial structure of cities is the substratum over which human urban life develops. Understanding the relevance of the streets that form such substratum is fundamental to understand the movements of people in the city. Our purpose in this study is to propose and explore methods to find relevant streets sets that conforms the basis of complex urban networks. We used two methods based on previous work by Volchenkov and Blanchard (2008) and Shetty and Adibi (2005), and a method developed by us. To illustrate the use of these tools, we performed an analysis of the main streets network of the coastal city of Ensenada, Mexico.
We introduce a method to estimate the complexity function of symbolic dynamical systems from a fi... more We introduce a method to estimate the complexity function of symbolic dynamical systems from a finite sequence of symbols. We test such complexity estimator on several symbolic dynamical systems whose complexity functions are known exactly. We use this technique to estimate the complexity function for genomes of several organisms under the assumption that a genome is a sequence produced by a (unknown) dynamical system. We show that the genome of several organisms share the property that their complexity functions behaves exponentially for words of small length ℓ (0 ≤ ℓ ≤ 10) and linearly for word lengths in the range 11 ≤ ℓ ≤ 50. It is also found that the species which are phylogenetically close each other have similar complexity functions calculated from a sample of their corresponding coding regions.
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.
We introduce a distance in the space of fully-supported probability measures on one-dimensional s... more We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to thed-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of g-measures, it allows to assess the continuity of the entropy at g-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite g-functions, to the preservation at the limit, of certain ergodic properties for the associate g-measures.
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.
Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-... more Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-invariant measures on Ω. We study the relation between the restriction In of I to the finite cubes {−n,. .. , n} d ⊂ Z d , and the polytope of "locally invariant" measures I loc n. We are especially interested in the geometry of the convex set In which turns out to be strikingly different when d = 1 and when d ≥ 2. A major role is played by shifts of finite type which are naturally identified with faces of In, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of In, although in dimension d ≥ 2 there are also extreme points which arise in other ways. We show that In = I loc n when d = 1, but in higher dimension they differ for n large enough. We also show that while in dimension one In are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of In for all large enough n. This work is part of the project 'CrystalDyn' funded by the Agence Nationale de la Recherche. M. Hochman is supported by NSF grant 0901534. The authors acknowledge L. Sadun and Robert F. Williams for allowing the reproduction of some figures from one of their papers.
Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-... more Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-invariant measures on Ω. We study the relation between the restriction In of I to the finite cubes {−n,. .. , n} d ⊂ Z d , and the polytope of "locally invariant" measures I loc n. We are especially interested in the geometry of the convex set In which turns out to be strikingly different when d = 1 and when d ≥ 2. A major role is played by shifts of finite type which are naturally identified with faces of In, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of In, although in dimension d ≥ 2 there are also extreme points which arise in other ways. We show that In = I loc n when d = 1, but in higher dimension they differ for n large enough. We also show that while in dimension one In are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of In for all large enough n. This work is part of the project 'CrystalDyn' funded by the Agence Nationale de la Recherche. M. Hochman is supported by NSF grant 0901534. The authors acknowledge L. Sadun and Robert F. Williams for allowing the reproduction of some figures from one of their papers.
We propose a geometric point of view to study the structure of ground states in lattice models, e... more We propose a geometric point of view to study the structure of ground states in lattice models, especially those with 'non-periodic long-range order' which can be seen as toy models for quasicrystals. In a lattice model, the configuration space is S Z d where S is a finite set, and Θ denotes action of the group Z d by translation or 'shift'. Given a shift-invariant potential Φ, ground states are none other than the shift-invariant probability measures supported on the set of ground configurations of that potential, i.e., those configurations with minimal specific energy. The ground states of Φ are supported by a multi-dimensional subshift of the d-dimensional 'full shift' (S Z d , Θ). This subshift may be minimal or not, uniquely ergodic or not, and with entropy zero or not. These three properties are different notions of order. For a finite-range potential, the subshift carrying its ground states configurations is a shift of finite type (SFT). The corresponding ground states are then naturally associated with the boundary a certain finite-dimensional convex polytope. This boundary becomes drastically different from d = 1 to d ≥ 2. This is because when d ≥ 2 there exist uniquely ergodic SFT's with no periodic configurations which can be seen as toy-models of quasicrystals. Here we construct such an example from the Penrose tiling. The framework we propose may help to investigate the stability of such models of quasicrystals when one perturbs the ad hoc potential one can always construct to have a given uniquely ergodic SFT as its ground state.
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).
R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshif... more 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, 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 to expose some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way of the possibilities of this correspondence. 1 σ-invariant means that σzX " X for all z P Z 2 R
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...
It is a current practice to organize biological data in a network structure where vertices repres... more It is a current practice to organize biological data in a network structure where vertices represent biological components and arrows represent their interactions. A great diversity of graph theoretical notions, such as clustering coefficient, network motifs, centrality, degree distribution, etc., have been developed in order to characterize the structure of these networks. However, none of the existent characterizations allow us to determine global similarity among networks of different sizes. It is the aim of the present paper to introduce a mathematical tool to compare networks not only with regard to their topological structure, but also in their dynamical capabilities. For this reason we aim to propose a pseudo-distance between networks, built around the notions of determination and dominancy, concepts recently introduced in the context of regulatory dynamics on networks. We use our proposed pseudo-distance to compare networks from the following bacteria: E. coli, B. subtilis, P. aeruginosa, M. tuberculosis, S. aureus and C. glutamicum. We also use this pseudo-distance to compare these real bacterial networks with equivalent homogeneous, scale-free and geometric three dimensional random networks. We found that even when bacterial networks are characterized with different levels of detail, have different sizes and represent different aspects of the organisms, the proposed pseudo-distance captures all these characteristics, and indicates how similar they are or not from random networks.
International Journal of Society Systems Science, 2014
Spatial structure of cities is the substratum over which human urban life develops. Understanding... more Spatial structure of cities is the substratum over which human urban life develops. Understanding the relevance of the streets that form such substratum is fundamental to understand the movements of people in the city. Our purpose in this study is to propose and explore methods to find relevant streets sets that conforms the basis of complex urban networks. We used two methods based on previous work by Volchenkov and Blanchard (2008) and Shetty and Adibi (2005), and a method developed by us. To illustrate the use of these tools, we performed an analysis of the main streets network of the coastal city of Ensenada, Mexico.
We introduce a method to estimate the complexity function of symbolic dynamical systems from a fi... more We introduce a method to estimate the complexity function of symbolic dynamical systems from a finite sequence of symbols. We test such complexity estimator on several symbolic dynamical systems whose complexity functions are known exactly. We use this technique to estimate the complexity function for genomes of several organisms under the assumption that a genome is a sequence produced by a (unknown) dynamical system. We show that the genome of several organisms share the property that their complexity functions behaves exponentially for words of small length ℓ (0 ≤ ℓ ≤ 10) and linearly for word lengths in the range 11 ≤ ℓ ≤ 50. It is also found that the species which are phylogenetically close each other have similar complexity functions calculated from a sample of their corresponding coding regions.
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.
We introduce a distance in the space of fully-supported probability measures on one-dimensional s... more We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to thed-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of g-measures, it allows to assess the continuity of the entropy at g-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite g-functions, to the preservation at the limit, of certain ergodic properties for the associate g-measures.
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.
Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-... more Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-invariant measures on Ω. We study the relation between the restriction In of I to the finite cubes {−n,. .. , n} d ⊂ Z d , and the polytope of "locally invariant" measures I loc n. We are especially interested in the geometry of the convex set In which turns out to be strikingly different when d = 1 and when d ≥ 2. A major role is played by shifts of finite type which are naturally identified with faces of In, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of In, although in dimension d ≥ 2 there are also extreme points which arise in other ways. We show that In = I loc n when d = 1, but in higher dimension they differ for n large enough. We also show that while in dimension one In are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of In for all large enough n. This work is part of the project 'CrystalDyn' funded by the Agence Nationale de la Recherche. M. Hochman is supported by NSF grant 0901534. The authors acknowledge L. Sadun and Robert F. Williams for allowing the reproduction of some figures from one of their papers.
Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-... more Let Σ be a finite alphabet, Ω = Σ Z d equipped with the shift action, and I the simplex of shift-invariant measures on Ω. We study the relation between the restriction In of I to the finite cubes {−n,. .. , n} d ⊂ Z d , and the polytope of "locally invariant" measures I loc n. We are especially interested in the geometry of the convex set In which turns out to be strikingly different when d = 1 and when d ≥ 2. A major role is played by shifts of finite type which are naturally identified with faces of In, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of In, although in dimension d ≥ 2 there are also extreme points which arise in other ways. We show that In = I loc n when d = 1, but in higher dimension they differ for n large enough. We also show that while in dimension one In are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of In for all large enough n. This work is part of the project 'CrystalDyn' funded by the Agence Nationale de la Recherche. M. Hochman is supported by NSF grant 0901534. The authors acknowledge L. Sadun and Robert F. Williams for allowing the reproduction of some figures from one of their papers.
We propose a geometric point of view to study the structure of ground states in lattice models, e... more We propose a geometric point of view to study the structure of ground states in lattice models, especially those with 'non-periodic long-range order' which can be seen as toy models for quasicrystals. In a lattice model, the configuration space is S Z d where S is a finite set, and Θ denotes action of the group Z d by translation or 'shift'. Given a shift-invariant potential Φ, ground states are none other than the shift-invariant probability measures supported on the set of ground configurations of that potential, i.e., those configurations with minimal specific energy. The ground states of Φ are supported by a multi-dimensional subshift of the d-dimensional 'full shift' (S Z d , Θ). This subshift may be minimal or not, uniquely ergodic or not, and with entropy zero or not. These three properties are different notions of order. For a finite-range potential, the subshift carrying its ground states configurations is a shift of finite type (SFT). The corresponding ground states are then naturally associated with the boundary a certain finite-dimensional convex polytope. This boundary becomes drastically different from d = 1 to d ≥ 2. This is because when d ≥ 2 there exist uniquely ergodic SFT's with no periodic configurations which can be seen as toy-models of quasicrystals. Here we construct such an example from the Penrose tiling. The framework we propose may help to investigate the stability of such models of quasicrystals when one perturbs the ad hoc potential one can always construct to have a given uniquely ergodic SFT as its ground state.
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