Mention the distinctive features of Vesara style and discuss how it developed under the Chalukyas and the Hoysalas describing the architecture of one temple of each dynasty.

The Vesara style has characteristics of both the Northern as well as Dravidian temple architecture. It has spire shaped structure on top known as the Shikhara. 
The salient features of this style are that it reduces the height of the temple however it retains the tiers.
The buildings of the Western Chalukyas are widely dispersed, which reflect the system of the local government and decentralization in the Western Chalukyan Administration.
The major improvement over the previous Badami Chalukya temple was the “Architectural Articulation” or ornamentation on the outer walls of the shrine. The presence of Figure sculpture such as Heroes of Ramayana and Mahabharata and loving couples (Mithuna) was additional structure of these temples at the earliest period. The Western Chalukyan Temples are either Ekakuta (one mandapa of one shrine) or Dvikuta (a common hall attached to two shrines). Mahadeva Temple, Itagi is a classic example of Vesara style by the Chalukyas.
The Hoyasala temple architecture was heavily influenced by the Western Chalukyas, Cholas as well as Pallavas, though there was a departure from the Chalukyan style.
This is evident from the fact that in the beginning, the temples were not over decorated, but the later temples have this feature in almost of its totality. Chennakesava Temple, Belur is the finest example under them.