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Discrete & Continuous Dynamical Systems - A
December 2012 , Volume 32 , Issue 12
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2012, 32(12): 4069-4110
doi: 10.3934/dcds.2012.32.4069
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Abstract:
We study existence and spectral stability of stationary solutions of the Dafermos regularization of a much-studied diffusive-dispersive equation with cubic flux. Our study includes stationary solutions that corresponds to Riemann solutions consisting of an undercompressive shock wave followed by a compressive shock wave. We use geometric singular perturbation theory (1) to construct the solutions, and (2) to show that asmptotically, there are no large eigenvalues, and any order-one eigenvalues must be near $-1$ or a certain number $\lambda^*$. We give numerical evidence that $\lambda^*$ is also $-1$. Finally, we use pseudoexponential dichotomies to show that in a space of exponentially decreasing functions, the essential spectrum is contained in Re$ \lambda \le -\delta <0 $.
We study existence and spectral stability of stationary solutions of the Dafermos regularization of a much-studied diffusive-dispersive equation with cubic flux. Our study includes stationary solutions that corresponds to Riemann solutions consisting of an undercompressive shock wave followed by a compressive shock wave. We use geometric singular perturbation theory (1) to construct the solutions, and (2) to show that asmptotically, there are no large eigenvalues, and any order-one eigenvalues must be near $-1$ or a certain number $\lambda^*$. We give numerical evidence that $\lambda^*$ is also $-1$. Finally, we use pseudoexponential dichotomies to show that in a space of exponentially decreasing functions, the essential spectrum is contained in Re$ \lambda \le -\delta <0 $.
2012, 32(12): 4111-4131
doi: 10.3934/dcds.2012.32.4111
+[Abstract](2328)
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For the dynamics defined by a sequence of bounded linear operators in a Banach space, we establish the robustness of the notion of exponential dichotomy. This means that an exponential dichotomy persists under sufficiently small linear perturbations. We consider the general cases of a nonuniform exponential dichotomy, which requires much less than a uniform exponential dichotomy, and of a noninvertible dynamics or, more precisely, of a dynamics that may not be invertible in the stable direction.
For the dynamics defined by a sequence of bounded linear operators in a Banach space, we establish the robustness of the notion of exponential dichotomy. This means that an exponential dichotomy persists under sufficiently small linear perturbations. We consider the general cases of a nonuniform exponential dichotomy, which requires much less than a uniform exponential dichotomy, and of a noninvertible dynamics or, more precisely, of a dynamics that may not be invertible in the stable direction.
2012, 32(12): 4133-4147
doi: 10.3934/dcds.2012.32.4133
+[Abstract](2133)
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Abstract:
In this paper we investigate ``double rotations'', i.e., interval translation maps that when considered on the circle, have just two intervals of continuity. Using the induction procedure described by Suzuki et al., we show that Lebesgue a.e. double rotation is of finite type, i.e., it reduces to an interval exchange transformation. However, the set of infinite type double rotations is shown to have Hausdorff dimension strictly between $2$ and $3$, and carries a natural induction-invariant measure. It is also shown that non-unique ergodicity of infinite type double rotations, although occurring, is a-typical with respect to every induction-invariant probability measure in parameter space.
In this paper we investigate ``double rotations'', i.e., interval translation maps that when considered on the circle, have just two intervals of continuity. Using the induction procedure described by Suzuki et al., we show that Lebesgue a.e. double rotation is of finite type, i.e., it reduces to an interval exchange transformation. However, the set of infinite type double rotations is shown to have Hausdorff dimension strictly between $2$ and $3$, and carries a natural induction-invariant measure. It is also shown that non-unique ergodicity of infinite type double rotations, although occurring, is a-typical with respect to every induction-invariant probability measure in parameter space.
2012, 32(12): 4149-4170
doi: 10.3934/dcds.2012.32.4149
+[Abstract](2714)
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Abstract:
We show that there exists a $C^\infty$ volume preserving diffeomorphism $P$ of a compact smooth Riemannian manifold $\mathcal{M}$ of dimension 4, which is close to the identity map and has nonzero Lyapunov exponents on an open and dense subset $\mathcal{G}$ of not full measure and has zero Lyapunov exponent on the complement of $\mathcal{G}$. Moreover, $P|\mathcal{G}$ has countably many disjoint open ergodic components.
We show that there exists a $C^\infty$ volume preserving diffeomorphism $P$ of a compact smooth Riemannian manifold $\mathcal{M}$ of dimension 4, which is close to the identity map and has nonzero Lyapunov exponents on an open and dense subset $\mathcal{G}$ of not full measure and has zero Lyapunov exponent on the complement of $\mathcal{G}$. Moreover, $P|\mathcal{G}$ has countably many disjoint open ergodic components.
2012, 32(12): 4171-4182
doi: 10.3934/dcds.2012.32.4171
+[Abstract](2118)
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Abstract:
In smectic-A liquid crystals, a unity director vector $\boldsymbol{n}$ appear modeling an average preferential direction of the molecules and also the normal vector of the layer configuration. In the E's model [5], the Ginzburg-Landau penalization related to the constraint $|\boldsymbol{n}|=1$ is considered and, assuming the constraint $\nabla\times \boldsymbol{n}=0$, $\boldsymbol{n}$ is replaced by the so-called layer variable $\varphi$ such that $\boldsymbol{n}=\nabla\varphi$.
In this paper, a double penalized problem is introduced related to a smectic-A liquid crystal flows, considering a Cahn-Hilliard system to model the behavior of $\boldsymbol{n}$. Then, the issue of the global in time behavior of solutions is attacked, including the proof of the convergence of the whole trajectory towards a unique equilibrium state.
In smectic-A liquid crystals, a unity director vector $\boldsymbol{n}$ appear modeling an average preferential direction of the molecules and also the normal vector of the layer configuration. In the E's model [5], the Ginzburg-Landau penalization related to the constraint $|\boldsymbol{n}|=1$ is considered and, assuming the constraint $\nabla\times \boldsymbol{n}=0$, $\boldsymbol{n}$ is replaced by the so-called layer variable $\varphi$ such that $\boldsymbol{n}=\nabla\varphi$.
In this paper, a double penalized problem is introduced related to a smectic-A liquid crystal flows, considering a Cahn-Hilliard system to model the behavior of $\boldsymbol{n}$. Then, the issue of the global in time behavior of solutions is attacked, including the proof of the convergence of the whole trajectory towards a unique equilibrium state.
2012, 32(12): 4183-4194
doi: 10.3934/dcds.2012.32.4183
+[Abstract](1814)
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We prove that for the set of Exact Magnetic Lagrangians the pro-perty “There exist finitely many static classes for every cohomology class" is generic. We also prove some dynamical consequences of this property.
We prove that for the set of Exact Magnetic Lagrangians the pro-perty “There exist finitely many static classes for every cohomology class" is generic. We also prove some dynamical consequences of this property.
2012, 32(12): 4195-4207
doi: 10.3934/dcds.2012.32.4195
+[Abstract](2498)
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Abstract:
We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.
We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.
2012, 32(12): 4209-4227
doi: 10.3934/dcds.2012.32.4209
+[Abstract](2265)
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Abstract:
We show existence of global conservative solutions of the Cauchy problem for the Camassa--Holm equation $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with nonvanishing and distinct spatial asymptotics.
We show existence of global conservative solutions of the Cauchy problem for the Camassa--Holm equation $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with nonvanishing and distinct spatial asymptotics.
2012, 32(12): 4229-4246
doi: 10.3934/dcds.2012.32.4229
+[Abstract](2666)
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Abstract:
In this paper, we study a conditional long-time stable fully discrete finite element scheme for a Ginzburg-Landau model for nematic liquid crystal flow. We also obtain its time asymptotic convergence (when number of time steps go to infinity, fixed time step and mesh size) towards a unique critical point of the elastic energy subject to the finite element subspace. Finally, we estimate some convergence rates towards this limit critical point. To prove convergence of the whole sequence, a Lojasiewicz type inequality is used.
Moreover, we extend these results to other schemes given in [3] and [10].
In this paper, we study a conditional long-time stable fully discrete finite element scheme for a Ginzburg-Landau model for nematic liquid crystal flow. We also obtain its time asymptotic convergence (when number of time steps go to infinity, fixed time step and mesh size) towards a unique critical point of the elastic energy subject to the finite element subspace. Finally, we estimate some convergence rates towards this limit critical point. To prove convergence of the whole sequence, a Lojasiewicz type inequality is used.
Moreover, we extend these results to other schemes given in [3] and [10].
2012, 32(12): 4247-4263
doi: 10.3934/dcds.2012.32.4247
+[Abstract](1987)
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Abstract:
In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. By variational methods and analytic techniques, the best constant corresponding to the system is investigated, and the existence and nonexistence of ground state solutions to the system are established.
In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. By variational methods and analytic techniques, the best constant corresponding to the system is investigated, and the existence and nonexistence of ground state solutions to the system are established.
2012, 32(12): 4265-4285
doi: 10.3934/dcds.2012.32.4265
+[Abstract](1952)
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We deal with a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove $\mathbf{L}^{\infty }-$time decay estimates of small solutions. We also discuss existence and nonexistence of wave operators.
We deal with a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove $\mathbf{L}^{\infty }-$time decay estimates of small solutions. We also discuss existence and nonexistence of wave operators.
2012, 32(12): 4287-4305
doi: 10.3934/dcds.2012.32.4287
+[Abstract](2513)
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Abstract:
In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.
In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.
2012, 32(12): 4307-4320
doi: 10.3934/dcds.2012.32.4307
+[Abstract](2738)
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Abstract:
We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{ll} u_{tt} - \Delta u + b(t)u_t=|u|^{\rho}, & (t,x) \in \mathbb{R}^+ \times \mathbb{R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x), & x \in \mathbb{R}^N. \end{array}\right. (*) $$ When $b(t)=b_0(t+1)^{-\beta}$ with $b_0>0$ and $-1 < \beta <1$ and $\int_{{\bf R}^N} u_i(x)\,dx >0\,(i=0,1)$, we show that the time-global solution of ($*$) does not exist provided that $1<\rho \leq \rho_F(N):= 1+2/N$ (Fujita exponent). On the other hand, when $\rho_F(N)<\rho<\frac{N+2}{[N-2]_+}:= \left\{ \begin{array}{ll} \infty & (N=1,2), \\ (N+2)/(N-2) & (N \ge 3), \end{array} \right.$ the small data global existence of solution has been recently proved in [K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that $0 \le \beta<1$. We can prove the small data global existence even if $-1<\beta<0$. Thus, we conclude that the Fujita exponent $\rho_F(N)$ is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109--114].
We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{ll} u_{tt} - \Delta u + b(t)u_t=|u|^{\rho}, & (t,x) \in \mathbb{R}^+ \times \mathbb{R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x), & x \in \mathbb{R}^N. \end{array}\right. (*) $$ When $b(t)=b_0(t+1)^{-\beta}$ with $b_0>0$ and $-1 < \beta <1$ and $\int_{{\bf R}^N} u_i(x)\,dx >0\,(i=0,1)$, we show that the time-global solution of ($*$) does not exist provided that $1<\rho \leq \rho_F(N):= 1+2/N$ (Fujita exponent). On the other hand, when $\rho_F(N)<\rho<\frac{N+2}{[N-2]_+}:= \left\{ \begin{array}{ll} \infty & (N=1,2), \\ (N+2)/(N-2) & (N \ge 3), \end{array} \right.$ the small data global existence of solution has been recently proved in [K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that $0 \le \beta<1$. We can prove the small data global existence even if $-1<\beta<0$. Thus, we conclude that the Fujita exponent $\rho_F(N)$ is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109--114].
2012, 32(12): 4321-4360
doi: 10.3934/dcds.2012.32.4321
+[Abstract](2410)
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We prove the existence of certain analytic invariant manifolds associated with fixed points of analytic symplectic and volume preserving diffeomorphisms. The manifolds we discuss are not defined in terms of either forward or backward asymptotic convergence to the fixed point, and are not required to be stable or unstable. Rather, the manifolds we consider are defined as being tangent to certain "mixed-stable" linear invariant subspaces of the differential (i.e linear subspace which are spanned by some combination of stable and unstable eigenvectors). Our method is constructive, but has to face small divisors. The small divisors are overcome via a quadratic convergent scheme which relies heavily on the geometry of the problem as well as assuming some Diophantine properties of the linearization restricted to the invariant subspace. The theorem proved has an a-posteriori format (i.e. given an approximate solution with good condition number, there is one exact solution close by). The method of proof also leads to efficient algorithms.
We prove the existence of certain analytic invariant manifolds associated with fixed points of analytic symplectic and volume preserving diffeomorphisms. The manifolds we discuss are not defined in terms of either forward or backward asymptotic convergence to the fixed point, and are not required to be stable or unstable. Rather, the manifolds we consider are defined as being tangent to certain "mixed-stable" linear invariant subspaces of the differential (i.e linear subspace which are spanned by some combination of stable and unstable eigenvectors). Our method is constructive, but has to face small divisors. The small divisors are overcome via a quadratic convergent scheme which relies heavily on the geometry of the problem as well as assuming some Diophantine properties of the linearization restricted to the invariant subspace. The theorem proved has an a-posteriori format (i.e. given an approximate solution with good condition number, there is one exact solution close by). The method of proof also leads to efficient algorithms.
2012, 32(12): 4361-4390
doi: 10.3934/dcds.2012.32.4361
+[Abstract](3101)
+[PDF](559.2KB)
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This paper presents a study of the nonlinear wave equation with $p$-Laplacian damping: \[ u_{tt} - \Delta u - \Delta _p u_t = f(u) \] evolving in a bounded domain $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent, i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H{1\atop 0}(\Omega)$ into $L^2(\Omega)$. The nonlinear term $- \Delta _p u_t $ acts as a strong damping where the $-\Delta _p$ denotes the $p$-Laplacian. Under suitable assumptions on the parameters and with careful analysis involving the Nehari Manifold, we prove the existence of a global solution and estimate the decay rates of the energy.
This paper presents a study of the nonlinear wave equation with $p$-Laplacian damping: \[ u_{tt} - \Delta u - \Delta _p u_t = f(u) \] evolving in a bounded domain $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent, i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H{1\atop 0}(\Omega)$ into $L^2(\Omega)$. The nonlinear term $- \Delta _p u_t $ acts as a strong damping where the $-\Delta _p$ denotes the $p$-Laplacian. Under suitable assumptions on the parameters and with careful analysis involving the Nehari Manifold, we prove the existence of a global solution and estimate the decay rates of the energy.
2012, 32(12): 4391-4407
doi: 10.3934/dcds.2012.32.4391
+[Abstract](2535)
+[PDF](452.2KB)
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We study the recurrence to mistake dynamical balls, that is, dynamical balls that admit some errors and whose proportion of errors decrease tends to zero with the length of the dynamical ball. We prove, under mild assumptions, that the measure-theoretic entropy coincides with the exponential growth rate of return times to mistake dynamical balls and that minimal return times to mistake dynamical balls grow linearly with respect to its length. Moreover we obtain averaged recurrence formula for subshifts of finite type and suspension semiflows. Applications include $\beta$-transformations, Axiom A flows and suspension semiflows of maps with a mild specification property. In particular we extend some results from [6, 10, 19] for mistake dynamical balls.
We study the recurrence to mistake dynamical balls, that is, dynamical balls that admit some errors and whose proportion of errors decrease tends to zero with the length of the dynamical ball. We prove, under mild assumptions, that the measure-theoretic entropy coincides with the exponential growth rate of return times to mistake dynamical balls and that minimal return times to mistake dynamical balls grow linearly with respect to its length. Moreover we obtain averaged recurrence formula for subshifts of finite type and suspension semiflows. Applications include $\beta$-transformations, Axiom A flows and suspension semiflows of maps with a mild specification property. In particular we extend some results from [6, 10, 19] for mistake dynamical balls.
2012, 32(12): 4409-4427
doi: 10.3934/dcds.2012.32.4409
+[Abstract](1868)
+[PDF](403.4KB)
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We extend the metric proof of the converse Lyapunov Theorem, given in [13] for continuous multivalued dynamics, by means of tools issued from weak KAM theory, to the case where the set-valued vector field is just upper semicontinuous. This generality is justified especially in view of application to discontinuous ordinary differential equations. The more relevant new point is that we introduce, to compensate the lack of continuity, a family of perturbed dynamics, obtained through internal approximation of the original one, and perform some stability analysis of it.
We extend the metric proof of the converse Lyapunov Theorem, given in [13] for continuous multivalued dynamics, by means of tools issued from weak KAM theory, to the case where the set-valued vector field is just upper semicontinuous. This generality is justified especially in view of application to discontinuous ordinary differential equations. The more relevant new point is that we introduce, to compensate the lack of continuity, a family of perturbed dynamics, obtained through internal approximation of the original one, and perform some stability analysis of it.
2012, 32(12): 4429-4443
doi: 10.3934/dcds.2012.32.4429
+[Abstract](1798)
+[PDF](384.9KB)
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We construct a sequence of generating functions $(h_n)_{n\in\mathbb{N}}$, arbitrarily close to an integrable system in the $C^r$ topology with $r<4$ for $n$ large enough. With the variational method, we prove that for a given rotation number $\omega$ and $n$ large enough, the exact monotone area-preserving twist maps generated by $(h_n)_{n\in\mathbb{N}}$ admit no invariant circles with rotation number $\omega$.
We construct a sequence of generating functions $(h_n)_{n\in\mathbb{N}}$, arbitrarily close to an integrable system in the $C^r$ topology with $r<4$ for $n$ large enough. With the variational method, we prove that for a given rotation number $\omega$ and $n$ large enough, the exact monotone area-preserving twist maps generated by $(h_n)_{n\in\mathbb{N}}$ admit no invariant circles with rotation number $\omega$.
2012, 32(12): 4445-4466
doi: 10.3934/dcds.2012.32.4445
+[Abstract](2065)
+[PDF](542.3KB)
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In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have $C^3$ smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.
In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have $C^3$ smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.
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