SRB measures for non-hyperbolic systems with multidimensional expansion

By José F. Alves. 2000

160 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/ceed27d5e51e19b7eac2e9a6dd02f526b2e20f6a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

Citations in the corpus

Markov structures and decay of correlations for non-uniformly expanding dynamical systems☆ (2002, methodology): “f is topologically mixing and has a unique ergodic absolutely continuous invariant (thus SRB) measure [1, 5]; 4.”
Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps (2008, background): “The weak hyperbolicity property of f is expressed through the notion of hyperbolic times, which was introduced in [1,3] for differentiable transformations.”
Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics (2006, background): “It was introduced by Alves in [6] (see also [8]) but some basic ideas behind this notion go back to the work of Pliss [204] and Mañé [174].”
RECURRENCE RATE IN RAPIDLY MIXING DYNAMICAL SYSTEMS (2004, methodology): “This includes for example Axiom A systems with equilibrium states, hyperbolic systems with singularities with their SBR measures such as those considered by Chernov in [7], many systems with a Young tower [17, 18], expanding maps with singularities such as in [14], some non-uniformly expanding maps ”
Equilibrium states for non-uniformly expanding maps (2003, background): “Now we need the notion of hyperbolic time, first introduced by Alves [ Alv00 ].”
Expanding Measures (2008, background): “(3) f |f (S1×I ) is strongly topologically transitive and has a unique ergodic absolutely continuous invariant (thus SRB) measure whose support is f (S1 × I ) [2]; (4) the Central Limit Theorem holds for f [5]; (5) the correlations of Hölder functions decay at least like e−c √ , for some c′ > 0 [31]”
Large Deviations for Non-Uniformly Expanding Maps (2006, methodology): “Such measures for non-uniformly hyperbolic maps where obtained more recently [46, 15, 16, 2].”
Physical measures for partially hyperbolic surface endomorphisms (2003, background): “Replacing/5 by/5oG -1, we can assume tha t /5 is supported on AC([1, 2]).”
Physical measures and absolute continuity for one-dimensional center direction (2013, background): “Following Pesin and Sinai [45] and Alves, Bonatti and Viana [6,21] (see also [19, Chapter 11]), we call Gibbs u-state any invariant probability measure m whose conditional probabilities (Rokhlin [52]) along strong unstable leaves are absolutely continuous with respect to the volume measure volu on t”
SRB measures for partially hyperbolic systems whose central direction is weakly expanding (2014, unknown): “…under condition (2) almost every point x has an inﬁnite sequence of hyperbolic times, while under (8), almost every point has an inﬁnite sequence of hyperbolic times and this sequence has positive density at inﬁnity , i.e. the statement of Lemma 4.2 holds with lim inf instead of lim sup [1, 4].”
Strong statistical stability of non-uniformly expanding maps (2003, result): “ introduced in [17] an open class of transformations in higher dimensions with non-uniformly expanding behavior for most of its orbits. The existence of SRB measures for Viana maps has been proved in [1]. Motivated by the results in [17] and [1], general conclusions on the existence of SRB measures ”
Martingale–coboundary decomposition for families of dynamical systems (2016, background): “namely T (x) = { x(1 + 2γxγ) x ∈ [0, 1 2) 2x− 1 x ∈ [12 , 1] .”