Unstable Topological Pressure for Partially Hyperbolic Diffeomorphisms with Sub-additive Potentials

Abstract

In this paper, we introduce the unstable topological pressure for C 1 -smooth partially hyperbolic diffeomorphisms with sub-additive potentials. Moreover, without any additional assumption, we have established the expected variational principle which connects this unstable topological pressure and the unstable measure theoretic entropy, as well as the corresponding Lyapunov exponent.

Sub-additive unstable topological pressure on saturated subsets

“…One can construct in a similar way a maximal (n, ) u-separated subset Γ as in Proposition 2.1 of [15], it is also an (n, ) u-spanning set, then for any n > M , one has…”

“…First, we construct a closed subset Z of G K such that CP u (f, G, Z) can arbitrarily close to P u (f, G). By the variational principle for unstable topological pressure in [15], for any η > 0, there exists µ ∈ M e f (M ) such that h u µ (f ) + G * (µ) > P u (f, G) − η 2 . For η above, by (1), there exists 0 > 0 and N ∈ N such that for any < 0 and n ≥ N , one has log γ n (G, ) n ≤ η 2 .…”

“…In [6], Hu, Wu, and Zhu investigated the unstable topological pressure for additive potentials. In [15] and [16], we worked on both unstable topological and measure theoretical pressure for C 1 -smooth partially hyperbolic diffeomorphisms with sub-additive potentials, the expected variational principle was established.…”