It is well known that all transitive uniformly expanding or hyperbolic dynamics have no phase transition with respect to H\"older continuous potentials. For more general dynamics, It is still an open question to classify all the dynamics having phase transition with respect to a certain class of regular potential. In dimension one, due to the work of Bomfim-Victor \cite{BC21}, it was proved that for all transitive $C^{1+\alpha}-$local diffeomorphism $f$ on the circle that is neither a uniformly expanding map nor invertible, has an unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\R \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except in a point $t_{0} \in (0 , 1]$. Furthermore, they proved spectral phase transitions, more specific, the transfer operator $\Lo_{f,-t\log|Df|}$ acting on the space of H\"older continuous functions, has the spectral gap property for all $t<t_0$ and does not have the spectral gap property for all $t\geq t_0$. We aim to prove similar results for two special cases of dynamics: a co-dimension $1$ partially hyperbolic endomorphism and transitive piecewise monotone on the circle. For the higher dimension, endomorphisms, we prove that thermodynamic and spectral phase transition lead to multifractal analysis of the Lyapunov spectrum, in particular we exhibit a class of partially hyperbolic endomorphism having phase transition with respect to the geometric potential in the central direction and describe the multifractal analysis of the central Lyapunov spectrum. For transitive piecewise monotone maps, we prove that the set of Hölder continuous potentials which doesn't have spectral and thermodynamic phase transition is dense in the uniform topology and the set of Hölder continuous potentials that has phase transition are not dense. Furthermore, we provide a description of phase transition based on the properties of the transfer operator and the type of convexity of the topological pressure function. In particular, we describe the behavior of the topological pressure function and the transfer operator associated.

It is known that all uniformly expanding dynamics have no phase transition with respect to H\"older continuous potentials. In this work we show that given a local diffeomorphism $f$ on the circle, that is neither a uniformly expanding dynamics nor invertible, the topological pressure function $\R \ni t \mapsto P_{top}(f,-t\log|Df|)$ is not analytical. In other words, $f$ has a thermodynamic phase transition with respect to geometric potential. Assuming that $f$ is transitive and that $Df$ is Hölder continuous, we show that there exists $t_0 \in (0,1]$ such that the transfer operator $\Lo_{f,-t\log|Df|}$ acting on the space of H\"older continuous functions, has the spectral gap property for all $t<t_0$ and has not the spectral gap property for all $t\geq t_0$. Similar results are also obtained when the transfer operator acts on the space of bounded variations functions and smooth functions. In particular, we show that in the transitive case $f$ has a unique thermodynamic phase transition and it occurs in $t_0$. In addition, if the loss of expansion of the dynamics occurs because of an indifferent fixed point or the dynamics admits an absolutely continuous invariant probability with positive Lyapunov exponent then $t_0=1$. As a consequence of thermodynamical and spectral phase transition, we obtain applications on multifractal analysis for the Lyapunov spectrum.

In this paper, we present some recent results in PI-theory about gradings and graded polynomial
identities for the algebra of triangular superior matrices of order 2 when it is defined as a Lie algebra.
More precisely, fixed a field K of characteristic 2 (finite or infinite), we present a classification of the
gradings on (UT2(K ),◦), the algebra of triangular superior matrices of order 2 over K endowed with
the product defined by x ◦ y = x y + yx. We also show generators for the TG-ideals of these gradings
as well give a positive answer to the Specht problem for the variety of Lie algebras generated by
UT2(K ) for each of these gradings.

We study the existence of solution and the asymptotic behavior for four specific models: wave equation with memory; coupled system of wave equations; coupled system of wave equation with heat equation (thermo-elastic system) and thermo-elastic system with external source. We analyze these problems under the interference of the monotono p-Laplacian operator. Our approach presents the recent aspects in the treatment of these models, in the scope of Partial Differential Equations. In this sense, we highlight our thermo-elastic model with p-Laplacian, as far as we know, it was presented for the first time in the literature precisely in this thesis. For the existence of a solution weuse the Faedo-Galerkin Method. For the analysis of the asymptotic behavior we used different andrecent techniques: in the stability analysis, the Theorem of M. Nakao and H. Kuwahara (1987) was used, and also the Theorem of P. Martinez (1999), based on a new inequality that generalizes the previous results of A. Haraux (1985) and M. Nakao (1978); for the finite time “Blow-up” analysis weused the Lemma de Y. Qin and J. Rivera (2004).

In this work we study one-dimensional expanding Lorenz maps f with the same singular point c. We show that if the orbits of singular values satisfy a condition of slow recurrence, then every ergodic invariant probability has slow recurrence to the singularity and it has finite Lyapunov exponent. Moreover, we show that generically the singular values do not belong to the basin of its SRB measure. Also, we show that singularity allows the existence of many ergodic invariant measures with full support, having positive entropy, fast recurrence to the singular region and infinite Lyapunov exponent. Furthermore, we consider a two-parameter standard family of these maps and prove that there is a cone in the parameter space, in which we find sets of points on the curves, which has positive Hausdorff dimension, so that the maps associated to these points have finite Lyapunov exponent for every ergodic invariant probability, and there is one and only one equilibrium state for a given H¨older potential.
Keywords: Expanding Lorenz, Lyapunov exponent, slow recurrence, two-parameter
family, Hausdorff dimension.

We consider the continuous time symmetric random walk with a slow bond on Z, which rates are equal to 1/2 for all bonds, except for the bond of vertices {−1, 0}, which associated rate is given by αn −β /2, where α > 0 and β ∈ [0, ∞] are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if β ∈ [0, 1), then it converges to the usual Brownian motion. If β ∈ (1, ∞], then it converges to the reflected Brownian motion. And at the critical value β = 1, it converges to the snapping out Brownian motion (SNOB) of parameter κ = 2α, which is a Brownian type-process recently constructed in 2016 by A. Lejay. We also provide Berry-Esseen estimates in the dual bounded Lipschitz metric for the weak convergence of one-dimensional distributions, which we believe to be sharp.

The aim of the present study was to study the following problems: Hydrodynamic Limit for SSEP with a slow membrane and Out of Balance Fluctuations for SSEP with a slow link. More precisely, the hydrodynamic boundary study model is the symmetric simple exclusion process (SSEP) in the d-dimensional torus, which has a membrane whose passage rate $? / (N ^?) $, $? > 0 $, is lower than the rate in other links. Due to the existence of this slow membrane, depending on the regimen of the parameter that regulates the slowness of this membrane, border conditions appear at macroscopic level. For $ \ beta \ in [0, 1) $, the hydrodynamic equation is given by the heat equation in the continuous torus, meaning that the slow membrane has no effect on the boundary. For $ \ in \ (1, \ infty) $, the hydrodynamic equation is given by the heat equation with Neumann edge conditions, meaning that the membrane divides the torus into two isolated regions. And, for the critical value $ \ beta = 1 $, the hydrodynamic equation is given by the heat equation with Robin boundary conditions, related to Fick's law. In the case of Fluctuations, the model under study is the one-dimensional SSEP that has a slow link. The great difficulty in the work of the Fluctuations was to obtain accurate estimates of transition probabilities of random walks of dimensions 1 and 2.

This paper provides some original contributions to the study of the relationships between graph amenity and Dynamic Systems. In the first main result of the work, we characterize graph amenity through the extension by graph of a complete Markov application, inspired by the works of Stadlbauer and Jaerisch for extension by group. We saw that under soft hypotheses, the graph is mild if, and only if, the spectral radius of the Transfer operator associated with graph extension is equal to 1. In addition, we have defined graph extension of Markov applications with embedded Gibbs Markov structure and non-uniformly expandable applications, and we show the graph amenity characterization through graph extension for these classes of dynamic systems. We conclude with two non-trivial applications. First, we show that Schreier's graph is mild, if the decay rate of the graph extension of a Markov application with embedded Gibbs-Markov structure is equal to 1, while in the other application, we consider an extension of a complete Markov application by a semigroup. In this scenario, the amenity of the semigroup is equivalent to the spectral radius of the Transfer operator equal to 1.

We study the rotation sets for homeomorphisms homotopic to the identity on the torus T d , d ≥ 2. In the conservative setting, we prove that there exists a Baire residual subset of the set Homeo0,λ(T 2 ) of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in T 2 , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that, for every d ≥ 2, the rotation set of C 0 -generic conservative homeomorphisms on T d is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property (on chain recurrent classes).