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This paper is the first of a series of three dedicated to a proof of theArnold diffusion conjecture for perturbations of {convex} integrableHamiltonian systems on $\mathbb{A}^3=\mathbb{T}^3\times \mathbb{R}^3$.  We consider systems of the form $H(\theta,r)=h(r)+f(\theta,r)$, where $h$ isa $C^\kappa$ strictly convex and superlinear function on $\mathbb{R}^3$ and$f\in C^\kappa(\mathbb{A}^3)$, $\kappa\geq2$. Given $e>\textrm{{Min}}\,h$ and afinite family of arbitrary open sets $O_i$ in $\mathbb{R}^3$ intersecting$h^{-1}(e)$, a diffusion orbit associated with these data is an orbit of $H$which intersects each open set $\widehat O_i=\mathbb{T}^3\timesO_i\subset\mathbb{A}^3$.  The first main result of this paper (Theorem I) states the existence (undercusp-generic conditions on $f$ in Mather's terminology) of "chains of compactand normally hyperbolic invariant $3$-dimensional cylinders" intersecting each$\widehat O_i$. Diffusion orbits drifting along these chains are then proved toexist in subsequent papers. The second main result (Theorem II) consists in aprecise description of the hyperbolic features of classical systems (sum of aquadratic kinetic energy and a potential) on$\mathbb{A}^2=\mathbb{T}^2\times\mathbb{R}^2$, which is a crucial step to proveTheorem I.

Chains of compact cylinders for cusp-generic nearly integrable convex systems on $\mathbb{A}^3$