For any continuous map f : M → M on a compact manifold M , we define the SRB-like probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f has SRB-like measures, even if SRB measures do not exist. We prove that the definition of SRB-like measures is optimal, provided that the purpose of the researcher is to describe the asymptotic statistics for Lebesgue almost every initial state. We prove that any isolated measure in the set O of SRBlike measures is SRB. Finally we conclude that if O is finite or countable infinite, then there exist (up to countable many) SRB measures such that the union of their basins cover M Lebesgue a.e. MSC2010: Primary 37A05; Secondary 28D05

For any C 1 diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesguealmost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating subbundle. As a consequence, if those exponents are non negative, and if the exponents on the dominated subbundle are non positive, those measures satisfy the Pesin Entropy Formula.

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 and transitivity properties. Our examples include transitive and recurrent attractors, that are either finite, countable, or a disjoint union of a Cantor set and a countable set. We also show that the attractor of a piecewise contracting map is usually a Lebesgue measure-zero set, and we give conditions ensuring that it is totally disconnected. Finally, we provide an example of piecewise contracting map with positive topological entropy and whose attractor is an interval.

We study the global dynamics of integrate and fire neural networks composed of an arbitrary number of identical neurons interacting by inhibition and excitation. We prove that if the interactions are strong enough, then the support of the stable asymptotic dynamics consists of limit cycles. We also find sufficient conditions for the synchronization of networks containing excitatory neurons. The proofs are based on the analysis of the equivalent dynamics of a piecewise continuous Poincaré map associated to the system. We show that for efficient interactions the Poincaré map is piecewise contractive. Using this contraction property, we prove that there exist a countable number of limit cycles attracting all the orbits dropping into the stable subset of the phase space. This result applies not only to the Poincaré map under study, but also to a wide class of general n-dimensional piecewise contractive maps.

We analyze the dynamics of a deterministic model of inhibitory neuronal networks proving that the discontinuities of the Poincaré map produce a never empty chaotic set, while its continuity pieces produce stable orbits. We classify the systems in three types: the almost everywhere (a.e.) chaotic, the a.e. stable, and the combined systems. The a.e. stable are periodic and chaos appears as bifurcations. We prove that a.e. stable systems exhibit limit cycles, attracting a.e. the orbits. Keywords: Chaotic sets, limit cycles, piecewise continuous dynamics, neuronal networks. INTRODUCTIONWe obtain rigorous mathematical results on the dynamics of a deterministic abstract discontinuous dynamical system, in a finite but large dimensional phase space. It comes from a non linear model of inhibitory neuronal networks, without delays, composed by equally or different 2 ≤ n < +∞ pacemaker neurons, evolving according to an autonomous differential equation in the inter-spike interval times, and interacting among them by synaptical instantaneous currents in the spiking instants. A vectorial autonomous differential equation governs the increasing potentials of the n neurons as a function on time t, only during the inter-spike regime. On the other hand, the spiking regime holds when at least one neuron, say i, reaches a threshold level and gives a spike. Due to the synaptic connections among the neurons, this spike produces sudden changes in the potentials of the other neurons j = i and resets the potential of the neuron i. The synapsis is assumed to be inhibitory, i.e. phase redeeming, meaning that the potentials of the receiving neurons suffer negative changes in the spiking times. That is why the inhibitory synapsis is modeled by a matrix {H ij } i =j , 1 ≤ i, j ≤ n of negative numbers H ij < 0 that represent the instantaneous discontinuity jumps in the potentials of the neurons j = i, produced by a spiking of the neuron i. The autonomous differential equation, verifying some very wide assumptions, governs the system during the inter-spike intervals of time. It leads to a Poincaré map (Sotomayor [1979]), which is contractive, as we prove in Theorem 4 (see also Budelli et al.[1996]).