Properties of invariant measures in dynamical systems with the shadowing property

Abstract

ABSTRACT. For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost 1-1 extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every c ≥ 0 and ε > 0 the collection of ergodic measures (supported on almost 1-1 extensions of odometers) with entropy between c and c + ε is dense in the space of invariant measures with entropy at least c. Moreover, if in addition the entropy function is upper semi-continuous, then for every c ≥ 0 ergodic measures with entropy c are generic in the space of invariant measures with entropy at least c.

On entropy of \Phi -irregular and \Phi -level sets in maps with the shadowing property

“…then R Φ (a, θ) ⊂ QR(t) and so we have h top (T, R Φ (a, θ)) ≤t. But for every x ∈ X and µ ∈ V T (x), by [15,Theorem 4.3] we can find a sequence of ergodic measures ν n with ν n → µ and h νn (T ) → h µ (T ). By the definition, Φdν n → Φdµ, and the support of every ergodic measure is internally chain recurrent, so:…”

“…Take any η > 0 and any invariant measure µ supported on some chain recurrent class in X with Φdµ ∈ (a − θ, a + θ). Recall that by [15,Theorem 4.3] there exists a sequence of ergodic measures ν n with ν n → µ, h νn (T ) → h µ (T ). and Φdν n → Φdµ.…”

“…In [17] and [18] the authors proved that both collections of uniformly recurrent points and regularly recurrent points are dense in the nonwandering set (that is, Toeplitz minimal systems are dense). Later, Li and Oprocha in [15] extended these results by showing that points whose orbit closures are odometers are dense in the non-wandering set and, moreover, in a transitive system with the shadowing property the collection of ergodic measures supported on odometers is dense in the space of invariant measures. They also proved that ergodic measures supported on Toeplitz systems may approximate other measures in the weak*-topology and in terms of value of entropy.…”

“…Successful approach in [15] provides motivation for the present work. We aim for better understanding of local dynamical structure of systems with the shadowing property.…”

Ergodic Optimization Restricted On Certain Subsets Of Invariant Measures

“…The concept of shadowing property (or known as the pseudo-orbit tracing property) were born during the fundamental work of Anosov and Bowen on Axiom A diffeomorphisms. In later researches, it was discovered that the concept can bring many interesting results (see for example [14,23,31,32]). Our work mainly base on the work in [19,23,27,38].…”