Existence of periodic orbits and horseshoes for mappings in a separable Banach space

Sub-additivity of measure-theoretic entropies of commuting transformations on Banach space

“…The condition (H3) is discussed in Sect. 4, it implies that the tangent space could be decomposition to the unstable, center and stable subspaces with respect to the transformations. For condition (H2) (ii), it is naturally satisfied when Df x (or Dg x ) is the sum of a compact operator and a contraction for each x ∈ A (see [8]), and will be used in Sect.…”

“…which called Lyapunov charts) with a Euclidean inner product, and so are the corresponding connect maps. The same method (Lyapunov charts) be constructed in [4] Sect. 3 on Banach space; (ii) in finite dimensions, we customary to consider the diffeomorphisms, for instance, the maps f, g in [1] be assumed as C 2 diffeomorphisms, so that, Lemma 4.3 and all the corollaries holds for f −1 , g −1 .…”

“…The unstable (stable) manifold theory is well known for finite-dimensional systems. For Hilbert space maps [3] and Banach space maps [4], stable and unstable manifolds were constructed using Lyapunov charts. Theorem 6.1 ([5] Theorem 6.1 unstable manifold theory).…”

Horseshoes and Lyapunov exponents for Banach cocycles over non-uniformly hyperbolic systems

“…For infinite-dimensional dynamical systems, Lian and Young [20, 21] generalized Katok’s results [16] to mappings and semiflows of Hilbert spaces. Recently, Lian and Ma [19] generalized Katok’s result to mappings of Banach spaces. For a Banach cocycle over a hyperbolic system f , Kalinin and Sadovskaya [14] proved that the upper and lower Lyapunov exponents of with respect to an ergodic measure can be approximated in terms of the norms of the values of on hyperbolic periodic points of f .…”

Characterization of SRB Measures for Random Dynamical Systems on Banach space

“…The nonuniform hyperbolic theory in infinite dimensional dynamical systems also be studied, Z. Lian, K. Lu [22] and M. Ghani Varzaneh, S. Riedel [17] construct the unstable and stable manifolds on random dynamical systems on Banach spaces. The more result about infinite dimensional dynamical systems can refer to [10,13,14,23,24,27].…”