In this paper, we get the dichotomy of fiber topological pressure of the fiber irregular set I_ψ(ω ) in the skew product systems with the fiber specification driven by an ergodic system. That is, for ℙ- a.e. ω∈Ω , I_ψ(ω ) is either empty or carries full fiber topological pressure. As an application, we prove that for a class of random conformal repellers with the fiber topological mixing property, the fiber irregular set has full fiber BS dimension and Hausdorff dimension.

Motivated by Helfter's notion of scaling, we introduce Bowen entropy scales and local measure-theoretic counterparts associated with general scaling families. Under a controlled decay condition, we establish a variational principle on compact subsets using Billingsley-type and Frostman-type arguments. We also define entropy and pressure scales via separated sets and prove an abstract variational principle using convex duality. The framework recovers classical entropy and includes examples related to slow entropy and entropy dimension. For the induced system, we give sufficient conditions under which the original system has zero upper capacity entropy scale if and only if the induced system does. Under a further condition on the scaling family, we prove that a positive upper capacity entropy scale of the original system implies an infinite upper capacity entropy scale of the induced system. We also provide a full-shift example showing that this conclusion does not hold for arbitrary scaling families. Finally, we define upper capacity entropy scales along prescribed increasing sequences of times and prove the corresponding zero-entropy result for induced systems.

In this paper, we investigate a broad class of systems for which Katok's intermediate entropy conjecture holds. In particular, we show that a dynamical system with an amenable group action satisfies Katok's intermediate entropy conjecture provided that it has the specification property and is asymptotically entropy expansive.

Based on the α -metric d_n^α(x,y)=max _0≤ i< ne^iαd(f^ix,f^iy), where α≥ 0 , we introduce α -estimation BS-Packing dimension and α -estimation packing topological pressure on subsets by the Carathéodory structure. Motivated by the classical Brin-Katok entropy, we define several measure-theoretic quantities and derive a variational principle for α -estimation BS-Packing dimension. We show that α -estimation BS-Packing dimension and α -estimation packing topological pressure are connected via Bowen’s equation. Additionally, we explore connections between α -estimation packing topological pressure and other α -pressure-like quantities.

Let (X, f) be a dynamical system possessing the specification property, and let φ be a continuous function. In this paper, we establish several conditional variational principles for the upper and lower Bowen/packing metric mean dimensions with potential, associated with the multifractal level set K_α := {x ∈ X: lim _n →∞1/n∑ _i=0^n-1φ (f^i x) = α} .

In this paper, we study typical periodic optimization (TPO) for almost additive potentials in two perturbation spaces. Our main setting is the Banach quotient ℰ_ orb(X,T) of orbit-Lipschitz almost additive potentials, where we extend the theory of maximizable sets and countable maximizable families developed by W. Huang, O. Jenkinson, L. Xu and Y. Zhang [Typical periodic optimization for dynamical systems: symbolic dynamics, Invent. Math. 245 (2026), 1–63], and establish a global structural theorem. For a countable maximizable family, global TPO holds if every non boundary member has X-extendable TPO and the boundary region has empty interior. As an application, we construct a compact system with global TPO for which ℰ_ orb(X,T) is infinite-dimensional and the maximizing periods in open locking regions are unbounded. For a fixed almost additive potential Φ, we also develop relative TPO theory on its Lipschitz leaf. When Φ=0, this framework reduces to classical Lipschitz TPO. We prove the corresponding leafwise structural theorem and give a non additive rank-one matrix example on a full shift.

For dynamical systems satisfying the approximate ℤ^d or ℤ_+^d-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In particular, we prove that the set of ergodic measures with any given intermediate entropy is generic in certain natural subspaces. As a consequence, this result confirms Katok's conjecture on the existence of ergodic measures with arbitrary intermediate entropy for such systems.

It is well-known that the relativized variational principle established by Bogenschutz and Kifer connects the fiber topological entropy and fiber measure-theoretic entropy. In context of random dynamical systems, metric mean dimension was introduced to characterize infinite fiber entropy systems. We give four types of measure-theoretic ε-entropies, called measure-theoretic entropy of partitions decreasing in diameter, Shapira's entropy, Katok's entropy and Brin-Katok local entropy, and establish four variational principles for metric mean dimension.

For infinite measure-theoretic entropy systems, we introduce the notion of measure-theoretic metric mean dimension of invariant measures for different types of measure-theoretic & varepsilon;-entropies, and show that measure-theoretic metric mean dimensions of different types of measure-theoretic & varepsilon;-entropies coincide with the packing metric mean dimension of the set of generic points of ergodic measures.

We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.

For an expansive homeomorphism, we investigate the relationship among dimension, entropy, and Lyapunov exponents. Motivated by Young's formula for surface diffeomorphisms, which links dimension and measure-theoretic entropy with hyperbolic ergodic measures, we construct the hyperbolic metric with two distinct Lyapunov exponents log b>0>-log a. We then examine the relationships between various types of entropies (entropy, r-neutralized entropy, and α-estimating entropy) and dimensions. We further prove the Eckmann-Ruelle Conjecture for expansive topological dynamical systems with hyperbolic metrics. Additionally, we establish variational principles for these entropy quantities.

Lindenstrauss and Tsukamoto in 2019 established double variational principle for mean dimension. In this paper, we focus on developing the mean dimension theory for ℤ^k -actions. Specifically, we establish a double variational principle for mean dimension of ℤ^k -actions for dynamical systems with the marker property.

. In this paper, we first prove the variational principle for amenable packing topological pressure. Then, we obtain an inequality concerning amenable packing pressure for factor maps. Finally, we show that the amenable packing topological pressure of the set of generic points for any invariant Borel probability measure mu, PP (G mu, {Fn}infinity n=1, f) is equal to the sum of the measuretheoretic entropy h mu (X, G) and f f d mu, when the system satisfies the almost specification property, or mu is ergodic.

Let f(i,) i = 1, 2, ... , k be continuous bundle random dynamical systems over an ergodic compact metric system (Omega, F, P, theta). Assume that a = (a(1), a(2), ... , a(k)) is an element of R-k with a(1) > 0 and a(i) > 0, f(i+1) is a factor of f(i) with a factor map pi(i) : E-i -> Ei+1. It is shown that the a-weighted Bowen topological entropy h(a)(omega, f(1), E-1, omega) is measurable in Omega, and denoted that h(a)(f(1), E-1) is the integration of h(a)(omega, f(1), E-1,E-omega) against P over Omega. We prove the following variational principle: h(a)(f(1), E-1) = sup{Sigma(k)(i=1) a(i)h(mu o tau-i-1-1)((r))(f(i))}, where the supremum is taken over the set of all mu is an element of M-P(1)(E-1, f(1)). In the case of random dynamical systems with an ergodic compact driving system, this gives an affirmative answer to the question posed by Feng and Huang [J. Math. Pures Appl. 106 (2016), 411-452] for fiber entropy theory. It also generalizes Kenyon-Peres' variational principle on compact random sets and provides a gaskets.

Let $(X,d)$ be a compact metric space, $f:X\rightarrow X$ be a continuous transformation with the specification property. we consider non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is empty or carries full Bowen upper and lower metric mean dimension.

Symbolic dynamical theory plays an important role in the research of amenability with a countable group. Motivated by the deep results of Dougall and Sharp, we study the group extensions for topologically mixing random shifts of finite type. For a countable group G, we consider the potential connections between relative Gurevič pressure (entropy), the spectral radius of random Perron-Frobenius operator and amenability of G. Given G^ ab by the abelianization of G where G^ ab=G/[G,G], we consider the random group extensions of random shifts of finite type between G and G^ ab. It can be proved that the relative Gurevič entropy of random group G extensions is equal to the relative Gurevič entropy of random group G^ ab extensions if and only if G is amenable. Moreover, we establish the relativized variational principle and discuss the unique equilibrium state for random group ℤ^d extensions.

In this paper, we prove that the weighted topological entropy defined by FK metric and Bowen metric are equal. Moreover, we establish a Brin-Katok formula and a Katok formula on weighted FK metric, respectively.

In this paper, we study the Feldman-Katok metric in random dynamical systems and establish corresponding fiber topological entropy formula, Brin-Katok local entropy formula and fiber Katok entropy formula by replacing Bowen metric with Feldman-Katok metric. It turns out that the Feldman-Katok metric is also the weakest metric that makes the entropy formulae valid on random dynamical systems.

Ovadia and Rodriguez-Hertz defined neutralized Bowen open ball as B_n(x,e^-nϵ)={y∈ X: d(T^jx, T^jy)