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In this work, it is presented a characterization of star property for a C 1 vector field based on Lyapunov functions. It is also obtained conditions to strong homogeneity for singular sets by using the notion of infinitesimal Lyapunov functions. As an application, we obtain some results related to singular hyperbolic sets for flows. [ABSTRACT FROM AUTHOR]

We show the existence and finiteness of physical (SRB) measures for a class of diffeomorphisms, each of which admits a dominated splitting T M = E c u ⊕ E c s such that any pseudo-physical measure admits only positive (resp. negative) Lyapunov exponents along E c u (resp. E c s). [ABSTRACT FROM AUTHOR]

We prove three formulas for computing the topological pressure of $ C^1 $ C 1 -generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there are no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For $ C^1 $ C 1 generic conservative diffeomorphism on compact surfaces with no dominated splitting and $ \phi _m(x):=-\frac {1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb {N} $ ϕ m (x) := − 1 m log ⁡ ‖ D x f m ‖ , m ∈ N , we show that there exist equilibrium states with zero entropy and there exists a transition point $ t_0 $ t 0 for the one parameter family $ \lbrace t \phi _m\rbrace _{t\geq ~0} $ { t ϕ m } t ≥ 0 , such that there is no equilibrium states for $ t \in [0, t_0) $ t ∈ [ 0 , t 0) and there is an equilibrium state for $ t \in [t_0,+\infty) $ t ∈ [ t 0 , + ∞). [ABSTRACT FROM AUTHOR]

We show that robustly transitive endomorphisms of a closed manifold must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result, we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation. [ABSTRACT FROM AUTHOR]

We extend Katok's result on 'the approximation of hyperbolic measures by horseshoes' to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M , preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$. We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles. [ABSTRACT FROM AUTHOR]

For typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal linear map. Using this result, we show that suitable assumptions on the periodic orbits have consequences over the entire subshift. [ABSTRACT FROM AUTHOR]

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$. [ABSTRACT FROM AUTHOR]

In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a C1-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow. [ABSTRACT FROM AUTHOR]

Introduction The influence of persistence behavior of a dynamical system on tangent bundle of a manifold is always a challenge in dynamical systems. Persistence properties have been studied on whole manifold or on some pieces with independent dynamics. Since shadowing property has an important role in the qualitative theory of dynamical systems, by focusing on various shadowing properties, such as usual shadowing, inverse shadowing, limit shadowing, many interesting results have been obtained. The notion of limit shadowing property introduced by S. Pilyugin who obtained its relation to other various shadowing. Blank introduced the notion of average-shadowing property. It is known that every Axiom A diffeomorphism restricted to a basic set has the average shadowing property. K. Sakai proved that the -interior of the set of all diffeomorphisms satisfying the average-shadowing property is characterized as the set of all Anosov diffeomorphisms. Asymptotic average shadowing (AASP) defined by R. Gu for continuous maps, combines to the limit shadowing property with the average shadowing property. Here we modify the notion (AASP) and define the limit average shadowing for diffeomorphisms (LASP). R. Gu presented some basic properties of the limit average shadowing for continuous maps. He proved that if a continuous map has the limit average shadowing on a compact metric space, then is chain transitive and that -hyperbolic homeomorphisms with limit average shadowing are topologically transitive. M. Kulczycki et.al., found some relations between LASP and the other notion of topological dynamics. They proved that a surjective map with specification property has the LASP. Also, they found the relation between LASP and shadowing property. They also have been shown that an expansive continuous map with shadowing property is LASP if and only if it is mixing. This paper follows the ideas of R. Gu and M. Kulczycki et.al. Here we define LASP for diffeomorphism with a slight modification of the continuous case. We give an example which shows that shadowing property and LASP are not equivalent. Also, we introduce the notion of - stably limit average shadowing for a closed -invariant subset of and show that if is -stably limit average shadowing and the minimal points of are dense there, then admits a dominated splitting. Statement of the results In this paper we give a system which has the limit average shadowing, but not the shadowing property. Also, one can give examples which have the shadowing but not the limit average shadowing property. Thus the limit average shadowing property does not imply the shadowing property. In fact, we can give a class of diffeomorphisms which have LASP, but not the shadowing property. In fact the following proposition gives a large class of diffeomorphisms satisfying the limit average shadowing. Proposition A. Let be a locally maximal -invariant set. If is the specific set for then is limit average shadowable. The main purpose of the paper is to characterize the closed -invariant set via limit average shadowing property in -open condition. So, we consider the notion of limit average shadowing property in geometric differential dynamical systems. First we show that if has the limit average shadowing property on a closed -invariant set then is chain transitive. By using chain transitivity and limit average shadowing property we can prove that is transitive. Proposition B. If is -stably limit average shadowable, then there is a neighborhood of and a neighborhood of such that contains neither almost sinks nor almost sources for any Since we have proved that if has the limit average shadowing property on a closed f-invariant set and minimal points of are dense then is transitive. It is essentially proved that under assumptions and conclusions of the Proposition B, admits a dominated splitting. Thus we get the main result of this paper. Theorem C. Let be a closed f-invariant set whose minimal points are dense there. If is -stably limit average shadowing then admits a dominated splitting../files/site1/files/62/2Abstract.pdf

We describe the structure of "substantially dissipative" complex Hénon maps admitting a dominated splitting on the Julia set. We prove that the Fatou set consists of only finitely many components, each either attracting or parabolic periodic. In particular, there are no wandering components and no rotation domains. Moreover, we show that J = J and the dynamics on J is hyperbolic away from parabolic cycles. [ABSTRACT FROM AUTHOR]

A vector field X is called a star flow if every periodic orbit of any vector field C¹-close to X is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3-or 4-manifold are either hyperbolic, or singular hyperbolic (see [MPP] for 3-manifolds and [LGW] for 4-manifolds). As it is defined, the notion of singular hyperbolicity forces the singularities in the same class to have the same index. However in higher dimensions (i.e. ≥ 5), [dL1] shows that singularities of different indices may be robustly in the same chain recurrence class of a star flow. Therefore the usual notion of singular hyperbolicity is not enough for characterizing the star flows. We present a form of hyperbolicity (called multisingular hyperbolicity) which makes the hyperbolic structure of regular orbits compatible with the one of singularities even if they have different indices. We show that multisingular hyperbolicity implies that the flow is star, and conversely we prove that there is a C¹-open and dense subset of the open set of star flows which are multisingular hyperbolic. More generally, for most of the hyperbolic structures (dominated splitting, partial hyperbolicity etc.) well defined on regular orbits, we propose a way of generalizing it to a compact set containing singular points. [ABSTRACT FROM AUTHOR]

We study bi-Lyapunov stable homoclinic classes for a C1 generic flow on a closed Riemannian manifold and prove that such a homoclinic class contains no singularity. This enables a parallel study of bi-Lyapunov stable dynamics for flows and for diffeomorphisms. For example, we can then show that a bi-Lyapunov stable homoclinic class for a C1 generic flow is hyperbolic if and only if all periodic orbits in the class have the same stable index. [ABSTRACT FROM AUTHOR]

This paper studies the M 0 -shadowing property for two types of volume-preserving diffeomorphisms defined on compact manifolds. For symplectic diffeomorphisms defined on symplectic manifolds, the C 1 -interior of the set of all symplectic diffeomorphisms with the M 0 -shadowing property is described by the set of the Anosov diffeomorphisms. If a volume-preserving diffeomorphism in Diff μ 1 (M) is a C 1 -stable M 0 -shadowing diffeomorphism, then M admits a volume-hyperbolic dominated splitting. [ABSTRACT FROM AUTHOR]

Let f be a C 1 -diffeomorphism which preserves an ergodic hyperbolic measure μ with positive measure-theoretic entropy. Assume that the Oseledec splitting T x M = E 1 (x) ⊕ ⋯ ⊕ E s (x) ⊕ E s + 1 (x) ⊕ ⋯ ⊕ E l (x) is dominated on the Oseledec basin Γ. We give extensions of Katok's Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes. Based on this, we also establish the continuity of the sub-additive topological pressure of the singular value potential. [ABSTRACT FROM AUTHOR]

For differentiable dynamical systems with dominated splittings, we give upper estimates on the measure-theoretic tail entropy in terms of Lyapunov exponents. As our primary application, we verify the upper semi-continuity of metric entropy in various settings with domination. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for C 1 -generic diffeomorphisms, if a homoclinic class contains periodic orbits of indices i and j with j > i + 1 , and the homoclinic class has no-domination of index l for any l ∈ { i + 1 , ... , j - 1 } , then there exists a non-hyperbolic ergodic measure with more than one vanishing Lyapunov exponents and whose support is the whole homoclinic class. Some other results are also obtained. [ABSTRACT FROM AUTHOR]

For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under C 1 perturbations and a global structures for the dynamics (such as hyperbolicity, partial hyperbolicity, dominated splitting). However, a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. In Bonatti (Star flows and multisingular hyperbolicity. arXiv:1705.05799, 2017) with Christan Bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities. Using this tool, we recover the results in Bonatti et al. (Ann Math 158(2):355–418, 2003) for flows, showing that robustly chain transitive sets have a weak form of hyperbolicity. allowing us to conclude as well the kind of hyperbolicity carried by the examples in Bonatti et al. (J Inst Math Jussieu 12(3):449–501, 2013) (a robust chain transitive singular attractor with periodic orbits of different indexes). Along with the results in [8], this shows that the way we propose to interpret the effect of singularities, has the potential to adapt to other settings in which there is coexistence of singularities and regular orbits with the goal of re-obtaining the results that we already know for diffeomorphisms. [ABSTRACT FROM AUTHOR]

We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Gu'eritaud-Guichard-Kassel-Wienhard [GGKW] and Kapovich-Leeb-Porti [KLP2] by different methods. We also give characterizations in terms of multicones and cone types inspired by the work of Avila-Bochi-Yoccoz [ABY] and Bochi-Gourmelon [BG]. Finally, we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti [KLP2]. [ABSTRACT FROM AUTHOR]

In dimension three and under certain regularity assumptions, we construct a renormalization scheme at the heterodimensional tangency of a nontransverse heterodimensional cycle associated with a pair of saddle-foci whose limit dynamics is a center-unstable Hénon-like family displaying blender-horseshoes. As a consequence, the initial cycle can be approximated in higher regularity topologies by diffeomorphisms having blender-horseshoes. [ABSTRACT FROM AUTHOR]

We say that a compact invariant set Λ of a C¹-vector field X on a compact boundaryless Riemannian manifold M is robustly shadowable if it is locally maximal with respect to a neighbourhood U of Λ, and there exists a C¹-neighbourhood of X such that for any, the continuation ΛY of Λ for Y and U is shadowable for Yt. In this paper, we prove that any chain transitive set of a C¹-vector field on M is hyperbolic if and only if it is robustly shadowable. [ABSTRACT FROM AUTHOR]

Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. Avila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting. [ABSTRACT FROM AUTHOR]

A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local C1-perturbations. Our main theorem shows that when one works in the C1-topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound following from Ruelle's inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller). This optimal entropy creation yields a number of consequences for C1- generic diffeomorphisms, especially in the absence of a dominated splitting. For instance, in the conservative settings, we find formulas for the topological entropy, deduce that the topological entropy is continuous but not locally constant at the generic diffeomorphism, and prove that these generic diffeomorphisms have no measure of maximum entropy. In the dissipative setting, we show the locally generic existence of infinitely many homoclinic classes with entropy bounded away from zero. [ABSTRACT FROM AUTHOR]

Metric entropies along a hierarchy of unstable foliations are investigated for C1 diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of robust hyperbolicity along periodic orbits homoclinically related to p (RNUHP) for conservative diffeomorphisms. It is proved that if f ∈ Diff1+m (M) is RNUHP, then f is Anosov. It is also shown that f admits a dominated splitting, provided that f is expansive conservative stable. [ABSTRACT FROM AUTHOR]

In this paper we prove a shadowing lemma for pseudo orbits made by quasi-hyperbolic strings. We allow singularities in question and hence, in particular, the quasi-hyperbolic strings are formulated by the rescaled linear Poincaré flow instead of the usual linear Poincaré flow. We also introduce the sectional Poincaré map and rescaled sectional Poincaré map for Lipschitz vector fields on Banach spaces in the article. [ABSTRACT FROM AUTHOR]

An important class of 'physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric 'push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems. [ABSTRACT FROM AUTHOR]

We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the -topology by positive matrices with an associated dominated splitting. [ABSTRACT FROM AUTHOR]

We consider a hyperbolic ergodic measure of a C 1 flow on a compact manifold. Under the hypothesis that there are no fixed points and that the Oseledec splitting of the normal bundle satisfies a limit domination property, we prove that the measure has a shadowing property. As an application of this result we prove that the measure can be approached on the weak ⁎ topology by measures supported on hyperbolic periodic orbits. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of Sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples. [ABSTRACT FROM AUTHOR]

This paper provides the Hausdorff dimension estimation for an ergodic hyperbolic measure of C¹-diffeomorphism on an m-dimensional compact Riemannian manifold with the assumption that its Oseledet's splitting is a dominated splitting. [ABSTRACT FROM AUTHOR]

We show that an invariant splitting for the tangent map to a smooth flow over a compact invariant subset is dominated if, and only if, the exterior power of the tangent map admits an invariant dominated splitting. For a C 1 vector field X on a 3-manifold, we obtain singular-hyperbolicity using only the tangent map DX of X and a family of indefinite and non-degenerate quadratic forms without using the associated flow X t and its derivative D X t . As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C 1 vector fields. [ABSTRACT FROM AUTHOR]

We prove that any C-stable weakly shadowable volume-preserving diffeomorphism de-fined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity. [ABSTRACT FROM AUTHOR]

We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative $$DX$$ of the vector field $$X$$ together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region. [ABSTRACT FROM AUTHOR]

Let H be an infinite dimensional Hilbert space, X a compact Hausdorff space and f : X → X a homeomorphism which preserves a Borel ergodic probability measure which is positive on non-empty open sets. We prove that non-uniformly Anosov cocycles are C0-dense in the family of partially hyperbolic cocycles with non-trivial unstable bundles. [ABSTRACT FROM AUTHOR]

In this paper we deal with an invariant ergodic hyperbolic measure μ for a diffeomorphism f, assuming that f is either C1+α or C1 and the Oseledec splitting of μ is dominated. We show that this system (f,μ) satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers. Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincaré recurrence, removing the assumption of the non-uniform specification property in the main theorem of "Recurrence and Lyapunov exponents" by Saussol, Troubetzkoy and Vaienti that establishes an inequality between Lyapunov exponents and local recurrence properties. Another consequence is the fact that any such measure is the weak limit of averages of Dirac measures at periodic points, as in a paper by Sigmund. One can show that the topological pressure can be calculated by considering the convenient weighted sums on periodic points whenever the dynamic is positive expansive and every measure with pressure close to the topological pressure is hyperbolic. [ABSTRACT FROM AUTHOR]

Abstract: Let p be a hyperbolic periodic point of a diffeomorphism f on a closed manifold M. Introducing here the notion of -persistently continuum-wise expansivity, we show that if the homoclinic class of f associated to p is -persistently continuum-wise expansive then (i) it admits a dominated splitting and (ii) it is hyperbolic provided it satisfies the chain condition. Moreover, we show that the chain recurrent set of f is -persistently continuum-wise expansive if and only if f satisfies both Axiom A and the no-cycle condition. [Copyright &y& Elsevier]

Let M be a closed smooth manifold M, and let f: M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f. We show that if f has the C-stably average shadowing property on Λ, then Λ admits a dominated splitting. [ABSTRACT FROM AUTHOR]

Abstract: We prove that a Hamiltonian star system, defined on a 2d-dimensional symplectic manifold M (), is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the 4-dimensional results in Bessa et al. (2010) [5]. [Copyright &y& Elsevier]

In this paper we introduce the notion of periodically expansiveness and discuss the homoclinic classes exhibit the property in a persistent way. More precisely, we prove that if a homoclinic class H( p, f) of a diffeomorphisms f is C-persistently periodically expansive then it admits a dominated splitting $${E \bigoplus F}$$ with dim ( E) = index ( p). We also prove that C-generically any locally maximal periodically expansive homoclinic class is hyperbolic. [ABSTRACT FROM AUTHOR]

Inspired by the work of Pujals and Sambarino on dominated splitting [E.R. Pujals and M. Sambarino, On the dynamics of dominated splitting, Ann. Math. 169 (2009), pp. 675–740], we present billiards with a modified reflection law which constitute simple examples of dynamical systems with limit sets having dominated splitting and where the dynamics is a rational or irrational rotation. [ABSTRACT FROM PUBLISHER]

A divergence-free vector field satisfies the star property if any divergence-free vector field in some C 1-neighbourhood has all singularities and all closed orbits hyperbolic. In this article, we prove that any divergence-free vector field defined on a Riemannian manifold and satisfying the star property is Anosov. It is also shown that a C 1-structurally stable divergence-free vector field is Anosov. Moreover, we prove that any divergence-free vector field can be C 1-approximated by an Anosov divergence-free vector field, or else by a divergence-free vector field exhibiting a heterodimensional cycle. [ABSTRACT FROM PUBLISHER]

For a homoclinic class H( p) of f ∈ Diff( M), f∣ H( p) is called R-robustly entropy-expansive if for g in a locally residual subset around f, the set Γ( x) = { y ∈ M: dist( g( x), g( y)) ≤ g3 (∀ n ∈ ℤ)} has zero topological entropy for each x ∈ H( p). We prove that there exists an open and dense set around f such that for every g in it, H( p) admits a dominated splitting of the form E ⊕ F ⊕ ... ⊕ F ⊕ G where all of F are one-dimensional and non-hyperbolic, which extends a result of Pacifico and Vieitez for robustly entropy-expansive diffeomorphisms. Some relevant consequences are also shown. [ABSTRACT FROM AUTHOR]

Let M be a closed manifold and f be a diffeomorphism on M. We show that if f has a nontrivial dominated splitting TM = E ⊕ F, then f cannot be minimal. The proof mainly uses Mañé's argument and Liao's selecting lemma. [ABSTRACT FROM AUTHOR]

Let ƒ be a diffeomorphism of a closed n-dimensional smooth manifold. In this paper, we show that if f has the C1-stably inverse shadowing property on a transitive set, then it admits a dominated splitting. [ABSTRACT FROM AUTHOR]

In this paper we show that there is a residual set R ⊂ Diff¹(M³) such that if f ∈ R is tame and has the average shadowing property, then f is Anosov or there is a non-hyperbolic ergodic measure with support on the whole of the manifold M. [ABSTRACT FROM AUTHOR]

Let p be a hyperbolic saddle of diffeomorphism f on closed manifold M and H(p, f) be the homoclinic class associated with it. In this article, we introduce the notion of C1-stably Lp shadowing and prove that if f is C1-stably Lp shadowable on a homoclinic class H(p, f) then, H(p, f) has a dominated splitting. Moreover, we prove that if f is C1-stably Lipschitz Lp shadowable on H(p, f) and H(p, f)-germ of f is expansive then the homoclinic class is hyperbolic. [ABSTRACT FROM AUTHOR]

Abstract: The highly complex nature of the transport in thermostatted billiards has been of interest in the last few decades because of industrial and medical applications. The onset of hyperbolic dynamics (deterministic chaos) in such a billiard has evidenced an interesting stabilization of the transport properties, especially in microporous media. Recently, different mathematical methods have been developed for establishing hyperbolicity in thermostatted billiards, among these, the Weyl-flow and the conformally symplectic structure techniques. This paper deals with analytical investigations on the possible hyperbolic nature of two thermostatted billiards: The nonequilibrium Ehrenfest gas (NEEG) and the pump model (PM). Despite numerical investigations supporting the idea of their dissipative dynamics, the hyperbolicity of these billiards has not been yet established. The analysis developed in this paper shows how the Weyl-flow technique has failed for NEEG, revealing the necessity to develop new strategies in order to obtain hyperbolicity. On the contrary, we prove that the PM has a conformally symplectic structure, which is the basis for establishing the hyperbolicity of such a hybrid dynamical system. [ABSTRACT FROM AUTHOR]

The article focuses on the C1-generic diffeomorphisms of the homoclinic class which features interior codimension one dominated splitting. It states that it is not recognized if the non-wandering set of generic diffeomorphism could show nonempty interior and not co-occur with the entire manifold. It discusses some of the mathematical solutions, theorems, and proofs for specific cases of diffeomorphisms with the results that could assist in finding the general solution for the problem.